observability analysis for power systems modeled at …
TRANSCRIPT
FEDERAL UNIVERSITY OF PARANÁ
ANDRÉ LUIZ LANGNER
OBSERVABILITY ANALYSIS FOR POWER SYSTEMS
MODELED AT THE SUBSTATION LEVEL INCLUDING
PMU
CURITIBA
2016
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ANDRÉ LUIZ LANGNER
OBSERVABILITY ANALYSIS FOR POWER SYSTEMS
MODELED AT THE SUBSTATION LEVEL INCLUDING
PMU
Dissertation submitted to the Graduate Program in Electrical Engineering from Federal University of Paraná, as partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering
Supervisor: Prof. Dr. Elizete Maria Lourenço
CURITIBA
2016
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Acknowledgements
First and foremost, I would like to express my deepest gratitude to my advisor,
Professor Elizete Maria Lourenço, for her guidance, support, and patience throughout
this M.S. study. It has been an honor and a great pleasure to be her research student.
I would like to extend my gratitude to Professors Djalma Mosqueira Falcão,
Thelma Solange Piazza Fernandes and Alexandre Rasi Aoki for serving as my
master’s thesis committee members.
I also would express my gratitude to Professor Ali Abur, from Northeastern
University, for his dedication with me during the time I have spent as visiting scholar at
his laboratory. I am indebted to express my gratitude to my friend Murat Gol, who also
contributed to this work.
Finally, I would like to give special thanks to my parents Edilson and Maristela
Langner, and my beloved wife Simone Pontarolo Langner, for their endless love and
support.
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RESUMO
O Estimador de Estados tem papel fundamental no monitoramento e controle em
tempo real de grandes sistemas de potência, sendo capaz de prover informações a
respeito do mais provável estado de operação da rede. Os primeiros algoritmos de
estimação de estados foram concebidos considerando o sistema inteiro modelado no
nível barra-ramo, uma vez que o processador de topologia reduz o sistema em um
modelo simplificado. Sendo assim, o presente trabalho foca na Estimação de Estados
Generalizada, na qual chaves e disjuntores são considerados no modelo da rede,
tendo seus respectivos estados estimados. A aplicação de medidas sincrofasoriais no
processo de Estimação de Estados e nas análises de observabildiade e criticidade de
medidas também fazem parte deste estudo. Neste trabalho são apresentadas duas
formulações para o uso de medidas sincrofasoriais na Estimação de Estados
Generalizada, bem como resultados que comprovam a aplicação das mesmas. Dentro
da proposta principal, está o desenvolvimento de um algoritmo numérico para análise
de observabilidade e criticidade de medidas em sistemas modelados no nível de
subestação. Testes são conduzidos no sistema de 14 barras dos IEEE, considerando
a modelagem explícita de algumas subestações, com diferentes configurações, e
simulando situações de falha na comunicação de medidas e do status dos disjuntores.
Os resultados mostram que os métodos implementados permitem a determinação da
observabildiade do sistema além da deteção de medidas e restrições críticas. Casos
de falha da método também são mostrados, bem como meios de mitigá-los. Uma
importante constação é sobre a vantagem do uso das medidas sincrofasoriais na
Estimação de Estados Generalizada, no qual sua aplicação elimina a criticidade de
restrições operacionais, as quais são crítitas devido a topologia da rede e não pela
quantidade e alocação de medidas.
Palavras chaves: Estimação de Estados. Generalizada. Análise de Observabilidade.
Medidas Sincrofasoriais.
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ABSTRACT
State Estimation (SE) plays a vital role in real-time monitoring and controlling of larger
power systems, as it provides the most likely operation state, and helps to keep it
working in a secure mode. The first SE algorithms were developed considering the
entire system modeled at the bus-branch level since the topology processor reduces
it in a simplified way. Having that in mind, this work is focused on the Generalized State
Estimation approach, in which switches and circuit breakers are considered in the
network model, with their status being estimated. The application of synchrophasor
measurements in the process of State Estimation, observability and measurement
criticality analyses, are also part of this study. This Master’s thesis presents two
formulations of using phasor measurements in the Generalized State Estimation, as
well as the results of such approaches, ensuring their application. The main proposal
is the development and deployment of a numerical algorithm to perform the
observability and criticality analyses in power systems modeled at the bus-section
level. Test are conducted over the IEEE 14 bus system considering the explicit
modeling of three substations, with different layouts, and simulating situations of
measurement and switch status failure. The results show that the deployed methods
allow the determination of the system observability besides the detection of critical
measurements and constraints. Cases where the method fails to provide desirable
results as also discussed, as well as ways of mitigating them. An important statement
regards the advantage of using synchrophasor measurements in the Generalized
State Estimation, in which their application eliminates critical operational constraints,
which are associated with the network topology, irrespectively of the measurement
quantity and allocation.
Keywords: Generalized State Estimation. Observability Analysis. Synchrophasor
Measurements.
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LIST OF FIGURES
Figure 1 – IEEE 14 Bus System – Bus-branch Model ............................................... 24
Figure 2 – IEEE 14 Bus System at the Substation Level ........................................... 32
Figure 3 – Closed Switch/Breaker from Bus 𝑘 to 𝑙 .................................................... 33
Figure 4 – Power flow measurements in a switching branch ..................................... 34
Figure 5 – Power Injection Measurement in a boundary bus .................................... 34
Figure 6 – Open Switch/Breaker from Bus 𝑘 to ......................................................... 35
Figure 7 – Substation model with phasor and conventional measurements .............. 40
Figure 8 – Power flow measurement on branch 𝑘 − 𝑚 .............................................. 48
Figure 9 – Power injection measurement at the generic bus 𝑡 .................................. 48
Figure 10 – 3 bus network ......................................................................................... 49
Figure 11 – Edge of a power flow measurement ....................................................... 51
Figure 12 – Edges of a power injection measurement .............................................. 52
Figure 13 – Measurement graph of 3 bus network .................................................... 52
Figure 14 – 6 bus network example .......................................................................... 53
Figure 15 – 6 bus network example, with indication of islands .................................. 57
Figure 16 – Example network at bus section level .................................................... 58
Figure 17 – Generalized measurement graph ........................................................... 60
Figure 18 – Bus and branches measured by a PMU ................................................. 66
Figure 19 – 3 bus / 5 nodes system – Example 1 ...................................................... 67
Figure 20 – Measurement graph of the 3 bus / 5 nodes system – Example 1 ........... 68
Figure 21 – 3 bus / 5 nodes system – Example 2 ...................................................... 70
Figure 22 – Measurement graph of the 3 bus / 5 nodes system disregarding PMU – First Step – Example 2 ......................................................................... 71
Figure 23 – Subsystem formed with Super-nodes – Example 2 ................................ 72
Figure 24 – Anchored and floating super-nodes – Example 2 ................................... 73
Figure 25 – Measurement graph of the 3 bus / 5 nodes system disregarding PMU –Effect of irrelevant injection measurement/constraint– Example 2 ....... 74
Figure 26 – 3 bus / 5 nodes system – Example 3 ...................................................... 75
Figure 27 – Subsystem formed with Super-nodes – Example 3 ................................ 75
Figure 28 – Subsystem formed with Super-nodes, considering the boundary injection – Example 3 ......................................................................................... 77
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Figure 29 – Algorithm Flowchart for Observability and Criticality Analysis for Measurements Type-2 ......................................................................... 79
Figure 30 – IEEE 14 Bus system with Modeled Substations ..................................... 81
Figure 31 – Substation modeled in detail – Base Case ............................................. 82
Figure 32 – Super-nodes of base case ..................................................................... 83
Figure 33 – Modeled Substations – Case Study A .................................................... 84
Figure 34 – Modeled Substations – Case Study B .................................................... 85
Figure 35 – Super-nodes of case study B ................................................................. 86
Figure 36 – Modeled Substations – Case Study C .................................................... 87
Figure 37 – Super-nodes of case study C ................................................................. 88
Figure 38 – Anchored and Floating super-nodes – Case C ...................................... 88
Figure 39 – Anchored and Floating super-nodes and injection measurement connections – Case C .......................................................................... 89
Figure 40 – Modeled Substation – Case Study D ...................................................... 90
Figure 41 – Super-nodes of case study D ................................................................. 90
Figure 42 – Super-nodes of case study D with irrelevant measurements ................. 91
Figure 43 – Super-node of case study D for Criticality Analysis ................................ 91
Figure 44 – Modeled substation – Case E ................................................................ 93
Figure 45 – Unified branch Model ........................................................................... 102
Figure 46 – Generic Bus ......................................................................................... 104
Figure 47 – IEEE 14 bus system modeled in PET ................................................... 107
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LIST OF ACRONYMS
CB - Circuit Breaker
EMS - Energy Management Systems
GPS - Global Positioning System
GSE - Generalized State Estimation
NTP - Network Topology Processor
OST - Observable Spanning Tree
PET - Power Education Toolbox
PMU - Phasor Measurement Unit
SE - State Estimation
TOA - Traditional Observability Analysis
TSE - Traditional State Estimation
TSO - Transmission System Operators
RTU - Remote Terminal Units
SCADA - Supervisory Control and Data Acquisition
WLS - Weighted Least Squares
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CONTENTS
1 INTRODUCTION .................................................................................................... 12
1.1 STATE OF THE ART REVIEW ............................................................................ 13
1.1.1 Observability Analysis ...................................................................................... 13
1.1.2 Generalized State Estimation ........................................................................... 17
1.1.3 Synchronized Phasor Measurement Units ....................................................... 18
1.2 DISSERTATION OBJECTIVES ........................................................................... 20
1.3 DISSERTATION OUTLINE .................................................................................. 21
2 POWER SYSTEM STATE ESTIMATION ............................................................... 23
2.1 TRADITIONAL STATE ESTIMATION .................................................................. 23
2.1.1 SE with Conventional Measurements ............................................................... 23
2.1.2 SE with Synchronized Phasor Measurements .................................................. 28
2.2 GENERALIZED STATE ESTIMATION ................................................................ 31
2.2.1 GSE with Conventional Measurements ............................................................ 33
2.2.2 GSE with Phasor Measurements ..................................................................... 39
2.3 SUMMARY OF THE CHAPTER .......................................................................... 45
3 OBSERVABILITY ANALYSIS ............................................................................... 46
3.1 TRADITIONAL OBSERVABILITY ANALYSIS: BUS-BRANCH ANALYSIS ......... 46
3.1.1 Network Observability....................................................................................... 47
3.1.1.1 – Basic Numerical Method ............................................................................. 50
3.1.1.2 – Basic Topological Method .......................................................................... 51
3.1.2 Identification of Observable Islands .................................................................. 53
3.2 GENERALIZED OBSERVABILITY ANALYSIS .................................................... 58
3.2.1 Generalized Network Observability .................................................................. 58
3.2.2 Determining Observable Islands in the Generalized Approach ........................ 61
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3.3 SUMMARY OF THE CHAPTER .......................................................................... 64
4 OBSERVABILITY AND CRITICALITY ANALYSIS FOR GENERALIZED STATE
ESTIMATION CONSIDERING PHASOR MEASUREMENTS .................................. 65
4.1 OBSERVABILITY AND CRITICALITY METHODS FOR MEASUREMENT CONFIGURATION TYPE-1 ............................................................................. 65
4.2 OBSERVABILITY AND CRITICALITY METHODS FOR MEASUREMENT CONFIGURATION TYPE -2 ............................................................................ 70
4.3 SUMMARY OF THE CHAPTER .......................................................................... 80
5 TESTS AND RESULTS ANALYSIS ...................................................................... 81
5.1 BASE CASE ........................................................................................................ 81
5.2 CASE A – MEASUREMENT CONFIGURATION TYPE 1 ................................... 83
5.3 CASE B – MEASUREMENT CONFIGURATION TYPE 2 ................................... 85
5.4 CASE C – MEASUREMENT CONFIGURATION TYPE 2 ................................... 87
5.5 CASE D – MEASUREMENT CONFIGURATION TYPE 2 ................................... 89
5.6 CASE E – ISLAND NODE ................................................................................... 93
5.7 SUMMARY OF THE CHAPTER .......................................................................... 94
6 CONCLUDING REMARK AND FUTURE STUDY ................................................. 95
6.1 CONCLUDING REMARKS .................................................................................. 95
6.2 FURTHER STUDY .............................................................................................. 96
REFERENCES .......................................................................................................... 97
APPENDIX A – POWER FLOW AND INJECTION EQUATIONS .......................... 102
APPENDIX B – SYSTEM DETAILS AND RESULTS OF GSE ALGORITHM ........ 107
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1 INTRODUCTION
Since the power systems have developed larger and more complex, and due
to the growing demand for reliability and security, the usage of real-time monitoring
and controlling of the entire system has become a necessity. Within that context, the
State Estimation plays a vital role as it gathers snap shots from remote terminal units
(RTU), via Supervisory Control and Data Acquisition (SCADA) system, which provides
visualization from the power plants to great load centers.
On account of it, the State Estimation method for Power System operation,
introduced by Fred Schweppe at the beginning of 70´s (SCHWEPPE, F.; WILDES,
1970), have been benefiting a great number of theoretical advances and practical
applications. Nowadays, SE is the backbone of modern Energy Management Systems
(EMS), and it provides the most likely state of operation.
Most of the commercial State Estimators (SE) adopt the so-called bus-branch
model of SE formulation. In such approach, a network topology processor (NTP)
reduces the system assuming correct information regarding switches and circuit
breaker status. Hence, it avoids the physical representation of switches and circuit
breakers, scaling down the size and complexity of the network.
In spite of the advantages of using the bus-branch model, there might occur
some drawbacks. For instance, it does not allow a detailed representation of
substations arrangements; switches and circuit breakers measurements may be lost,
as well as topology errors cannot be detectable.
To cope with such problems, Monticelli and Garcia (1991) proposed a new way
of modeling switches and breakers in order to have more information acquired from
SE algorithms, allowing the processing of topology erros. With the Generalized State
Estimation (GSE) it was possible not only estimate the conventional states (voltage
phasors of all buses) but also the switches and circuit breakers status, as much as the
power flow through them.
More recently, the advent of synchronized phasor measurements has also
aided the power system operation area, since such devices provide timestamp in
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Global Positioning Systems (GPS) synchronized measurements with a high accuracy
and precision.
These being said, the main motivation of this work is to explore the use of
synchronized phasor measurements in the Generalized State Estimation, as such field
of research has not been fully studied yet. Although other works have demonstrated
that Phasor Measurement Units (PMU) can benefit the observability and criticality
analysis, most of them focus in power systems modeled at the bus-branch level.
Therefore, there is a lot to be explored when the modeling of switches and circuit
breakers comes out.
This first chapter presents a state of the review, by briefly showing the
advances in the area of Power System State Estimation, the evolution of Observability
methods as well as the advent of synchronized phasor measurements. Along with that,
it also depicts the objectives and an outline of this work.
1.1 STATE OF THE ART REVIEW
The State Estimation technique was first introduced in the Power System area
by Schweppe in the 70’s, in a series of three papers that presented the Exact Model
(SCHWEPPE, F.; WILDES, 1970), the approximate model (SCHWEPPE; ROM, 1970),
and implementation issues regarding computational limitations (SCHWEPPE, 1970).
Although the problem of observability had not been formally recognized in the former
papers, the authors addressed questions about the meter placement in the estimator
performance.
1.1.1 Observability Analysis
Observability issues started to gain more attention after 1973 when many
researchers addressed efforts to consolidate methods to perform such analyses. In the
80’s, selected research groups published a variety of papers establishing
methodologies to determine the system observability. They proposed algorithms to
determine the system observability, and along with those when the network is found
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unobservable, methods of finding observable islands and placing measurements to
restate the system observability.
One of these research groups, compounded by Krumpholz, Clements and
Davis, focused on the Topological methodology to assess the system observability.
One of their papers (KRUMPHOLZ; CLEMENTS; DAVIS, 1980), discloses a practical
algorithm using the network topology to find a full rank spanning tree, which renders
the system observable. The algorithm finds such a spanning tree using a combinatorial
method and the graphs theory, which firstly processes the lines flow measurements.
Then, it processes the boundary injections, aiming at finding the so-called spanning
three, in an iterative manner. Besides evaluating the network observability, the
algorithm also identifies observable islands, for unobservable systems, and it makes
use of pseudo measurements to make the system observable.
In another paper, Clements, Davis and Krumpholz, (1981) emphasized the
problem of identification of critical measurements, which directly affects the detection
and identification of Bad Data. Other papers from this group disclosed modified
algorithms as a means to deal with measurement deficiency (CLEMENTS;
KRUMPHOLZ; DAVIS, 1982) so as to find maximal observable sub networks, as well
as to place measurements to recover the network observability (CLEMENTS;
KRUMPHOLZ; DAVIS, 1983).
Quintana, Simões Costa and Mandel (1982) propose a method to determine
the network observability through a Topological approach. In this paper, they used the
Graph Theory approach over Matroid Intersections. In the first place, the algorithm
finds an observable spanning tree processing the flow measurements, and so it does
with the injection measurements afterwards, by making use of a color scheme. They
also present the method results through tests in a realistic model of the Brazilian Power
System with 121 buses. In addition to that, the same researchers have also
investigated the critical measurements and the detectability of measurements errors
over their proposed algorithm (SIMÕES COSTA; PIAZZA; MANDEL, 1990).
The work of Nucera and Gilles (1991) have also focused on the Topological
approach by employing an optimal combinatorial algorithm. They compared their
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developments to the Krumpholz; Clements and Davis (1980)’ algorithms, which
revealed good advances in the computing process time.
Back in the 80’s, other researchers laid efforts on another approach to carry
out observability analysis. Monticelli and Wu have worked with the numerical methods,
and in two papers they explained the methodology used (MONTICELLI; WU, 1985a,
1985b). The first paper presents a complete theory about observability, with definitions,
theorems and proofs of the determination of network observability, unobservable
states, and identification of observable islands. The second one depicts the
deployment of the algorithms to determine the system observability and to identify
observable islands, by using the Jacobian and Gain matrix of the measurements. The
algorithms are iterative and both discuss the effects of irrelevant injection
measurements. A third paper from Monticelli and Wu proposed the orthogonal
transformations, as a means to circumvent ill-conditioning problems faced by
numerical methods (MONTICELLI; WU, 1986).
Falcão and Arias (1994) describe a numerical method through the factorization
of the linearized models in the echelon form, representing an evolution of the least
absolute value state estimation method. Along with that, they present a discussion
regarding critical measurements and Bad Data processing. Expósito, Abur and Ramos
(1995) investigate the use of loop equations as an alternative to traditional formulation
of the State Estimation, as well for observability purposes. They also explore the use
of current measurements (more abundant in distribution networks) which causes
multiple solutions due to the unlikelihood of determining the current direction (ABUR;
EXPÓSITO, 1997). After that, they use the loop equations to determine the network
observability regarding current measurements (EXPÓSITO; ABUR, 1998).
More recently, Gou and Abur (2000) offer a direct method to carry out the
observability analysis. The method consists of performing the triangular factorization
of the Gain matrix, manipulating the resulting lower and diagonal matrices by making
use of a numerical approach. The above-mentioned technique presents advantages
as it does not require the elimination of both irrelevant branches and irrelevant injection
measurements. The authors also suggest an algorithm for pseudo measurements
placement in order to restore the system observability in an iterative manner. As a
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matter of fact, in a second paper they propose an improved pseudo measurements
placement algorithm through a direct way (GOU; ABUR, 2001). Gou (2006) also
suggests a method for observability analysis based on the direct use of the Jacobian
matrix.
London, Alberto and Bretas (2007) introduce a new tool for assessing
measurement sets in the light of network observability, restoration, and identification
of critical measurements and critical sets. The method finds the critical information
using only the network-topology data over the concepts of the 𝐻∆ matrix, which is
processed by triangular factorization of the Jacobian matrix. Benedito et al. (2008) also
use concepts of the 𝐻∆ matrix, though for purposes of observability and identification
of observable islands, based on path graphs.
Almeida, Asada and Garcia (2008) disclose a direct numerical method for
observability analysis based on Gram matrix factorization, along with another method
to identify observable islands based on minimum norm solutions. They argue that the
method is easy of deploying thanks to its use for information already in State Estimation
(SE) routines, as well as for its capability of dealing with irrelevant measurements and
detecting observable islands. Another numerical method to determine the network
observability and identify observable islands, based on a numerical approach was
proposed by Silva, Simões Costa and Lourenço (2011) in which it uses orthogonal
Givens rotation. The methodology does not account Gain matrix since it operates
directly on the Jacobian matrix.
In summary, the Topological methods have advantages due to the fact that
they do not use floating point calculations, what may cause round-off errors (NUCERA;
GILLES, 1991). On the other hand, Numerical methods are easy to deploy as they
allow the employment of an already existing subroutines in a State Estimation program
(MONTICELLI; WU, 1985b). Aiming at taking advantage of the aforementioned
approaches, Korres and Katsikas (2003) introduce an hybrid method for observability
analysis. In short, the method firstly processes the flow measurements based on a
topological approach, forming the observable islands. After that, the boundary injection
measurements are retained for numerical analysis. Furthermore, in another paper,
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they illustrate a numerical method for topological observability analysis, using concepts
of the graph theory and echelon form (KORRES et al., 2003).
1.1.2 Generalized State Estimation
All the foregoing methodologies were proposed considering the power system
modeled at the so-called bus-branch level, in which the substations are modeled by
buses and transmission lines and transformers by their equivalent PI model. According
to Abur and Exposito (2004) the topology processor converts a bus section/switch
detailed model into a compact bus-branch model. In other words, it determines the
simplified model of the power system through the available data of measurements and
circuit breaker (CB) statuses.
Irving and Sterling (1982) were the first to investigate the use of substation
data for purposes of measurement error detection and correction, and Monticelli and
Garcia (1991) propose a new approach to run State Estimation algorithms in networks
modeled at the bus-section level. Their method allows the exact model of zero
impedance branches, as it applies the power flows through circuit breakers as new
state variables. Monticelli also published two more papers focusing on this approach
(MONTICELLI, 1993a, 1993b), setting the basis of the so-called Generalized State
Estimation (GSE) (ALSAC et al., 1998).
The extension of the numerical observability analysis was addressed by
Monticelli (1993b), in which the new state variables (the power flows through circuit
breakers) are represented by extra columns in the measurement Jacobian matrix. The
rank determination also provides information regarding the network observability. The
observable islands are found by means of the same approach of Monticelli and Wu
(1985b). Katsikas and Korres also worked with the numerical approach, unfolding a
direct method (KATSIKAS; KORRES, 2003), along with a simplified model, whose
purpose is to reduce the computation burden (KORRES; KATSIKAS, 2005).
The observability topological approach for systems modeled at the substation
level (bus section/switch level) was investigated by Simões Costa, Lourenço and
Clements (2002), as a means of extending the conventional method of graph theory.
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Such method includes power flows through circuit breakers (switching branches) as
new state variables, aiming at finding an observable spanning tree of full rank. In
addition to it, the concept used to find critical measurements and critical constraints
was extended.
1.1.3 Synchronized Phasor Measurement Units
The advent of GPS (Global Positioning System) synchronized measurements,
along with the use of microprocessors into substations has allowed measuring positive
sequence voltage phasors and positive sequence current phasors (PHADKE; THORP,
1986). Also known as PMU (Phasor Measurement Units), had its origin in the
development of a Symmetrical Component Distance Relay (SCDR) in the 70´s for
protection purposes (PHADKE; IBRAHIM; HLIBKA, 1977). Since then, it has enhanced
the state estimation performance (THORP; PHADKE; KARIMI, 1985).
Thorp, Phadke and Karimi (1985) present the concepts of using synchronized
measurements for state estimation purposes. It has been demonstrated that the
capability of directly measuring a state, the voltage angle, was able to improve the
convergence rates of the existing algorithms. Data reduction feature has also been
reported, in which the flow measurements are replaced by angle measurements.
However, such strategy has proven jeopardize the rejection of bad data. In another
paper, the same authors depict an algorithm that incorporates phasor measurements
in the state estimation problem (PHADKE; THORP, 1986).
Many researchers have presented different methodologies to use
synchronized phasor measurements in conjunction with SCADA measurements in
state estimation algorithms. Zhou et al. (2006) propose the use of phasor
measurements in a post processing linear estimator. In this approach, the
SCADA/conventional measurements are processed by a non-linear estimator, and the
phasor measurements processed afterward, with the results of the first stage. Nuqui
and Phadke (2007) have worked in the same way, presenting a hybrid linear state
estimator. Manousakis et al. (2013), in turn, propose to process the phasor
measurements first, in a linear estimator, and use the estimates as measurements with
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high weights, or equality constraints, in a non-linear estimator. Another approach is to
use SCADA/conventional measurements in conjunction with phasor measurements;
converting current phasor measurements into power flow ones is also an alternative
(ATANACKOVIC et al., 2008).
Zhu and Abur (2007) investigate the effects of choosing a reference bus angle
in the presence of PMU. They pointed out that if the slack bus has no PMU placed, it
may cause inconsistencies during the state estimation process. On the other hand, if
the slack bus has a PMU placed, it will have to provide accurate measurements.
Otherwise, errors will not be detectable biasing the final estimate. Their proposal does
not choose a reference bus in the presence of PMU, a fact that provides better results
for bad data detectability purposes.
In the light of observability analysis, many researchers suggested techniques
for optimal placement of PMU, in order to render a given power network fully
observable with a minimum number of measurements. Baldwin et al. (1993) and Nuqui
and Phadke (2005) have worked with the topological approach and a simulated
annealing technique to find optimal measurement design. Xu and Abur (2004) use the
numerical approach with the intent of finding an optimal meter placement, over an
integer programming technique and so did Chen and Abur (2006), but for purposes of
bad data detection. Koutsoukis et al. (2013) used a Recursive Tabu Search method
for optimal placement.
Out of the optimal placement techniques, London et al. (2009) propose the use
of the 𝐻∆ for redundancy and observability purposes, regarding conventional and PMU
measurements, and Korres and Manousakis (2012) use a hybrid algorithm for
observability checking and restoration.
The majority of the methodologies that made usage of synchronized phasor
measurements for purposes of SE and observability analysis have been carried out
considering the PMU capable of measuring all the adjacent lines of a given bus, which
means, unlimited channel numbers. However, the existing PMU come with a limited
number of channels, as recognized by Korkali and Abur (2009), who proposed an
optimal placement approach regarding this limitation. In the same way, Emami and
Abur (2010) suggest a robust measurement design considering the PMU capable of
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measuring the voltage phasor of a given bus, and only one adjacent line, called “branch
PMU”.
Another possible drawback, such as the availability of only synchronized
current phasor measurements was investigated by Gol and Abur (2013). They propose
a new methodology to include voltage magnitude measurements in the observability
and criticality analysis, in which PMU can only provide current phasor measurements.
The method handles an incidence matrix, which relates the states to measurements
by using the reduced echelon form for observability purposes, as well as to form a
sensitivity matrix to find critical measurements.
Regarding modern substations, the use of PMU inside of it has been
investigated by Jaén, Romero and Expósito (2005) whose work attempts to thoroughly
measure the substation through intelligent electronic devices (IED). Such devices are
effective for providing voltage and current magnitude, as well as angles. In this paper,
a three-phase Generalized State Estimator (GSE) was recommended for validation of
substation data.
On its turn, Yang, Sun and Bose (2011a, 2011b) also came up with a relevant
paper using phasor measurements, in the context of substation level SE. It firstly
processes current phasor measurements pondering current in CBs as states, and it
aims at identifying CB status errors.
1.2 DISSERTATION OBJECTIVES
This Master’s work focus on the research field of Power System State
Estimation, aiming to develop new techniques for real time modeling and analysis. This
field involves studies of topology and measurements errors, as well as observability
analysis in the generalized approach, considering the explicit modeling of switches and
circuit breakers. Since the synchronized phasor measurements units have become a
reality in power systems, and its benefits can boost the state estimation algorithms, the
application of such devices for State Estimation purposes are also one of the major
points of the research.
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In the light of it, the main objective here is to further study the use of
synchronized phasor measurements for observability and measurement criticality
purposes, considering the power system modeled at the bus-section level. It also aims
to propose a new technique to carry out such analyses considering modern methods,
already applied to systems modeled at the bus-branch model.
The specific objectives are the following:
To evaluate the main methods of observability analysis and computational
algorithms, such as topological and numerical ones;
To implement phasor measurements on the already developed Generalized
State Estimator algorithm;
To develop a suitable algorithm to provide observability and measurement
criticality analysis in power system modeled at the bus-section level;
To validate the developed algorithm over an IEEE benchmark test system.
1.3 DISSERTATION OUTLINE
This dissertation comprises six chapters and it is organized as follow: in the
current chapter, it is presented an introduction of this work, a literature review,
motivations for conducting such a research, and an outline regarding its contributions.
In the succeeding chapter, the methodology of State Estimation for power
systems modeled at the bus-branch level, as well as for power systems modeled at
the bus-section level is depicted. Furthermore, it also discusses the use of phasor
measurements, and its implications.
Chapter 3 covers the methodology related to the Observability analysis.
Consequently, the numerical approach is presented in details, as it is the method
adopted in this work. A brief review of the topological approach is as well introduced.
In addition, this chapter also presents both observability approaches for power systems
modeled at the bus-branch level and bus-section level. Chapter 4 acknowledges the
method of Observability and Criticality analysis for power systems modeled at the bus-
section level, emphasizing the use of phasor measurements. Moreover, the method is
22
presented with tutorial examples in a small system and an algorithm summarizes the
process at the end.
Chapter 5 reveals the results of the proposed method in the well-known IEEE
14 bus system, bearing in mind some substations modeled in detail. The test cases
illustrate possible situations for system operators, and the results show the advantages
of using phasor measurements inside the substations. Finally, in chapter 6, the main
contributions are outlined and discussed, as well as possible further studies.
23
2 POWER SYSTEM STATE ESTIMATION
This chapter aims at presenting the concepts of the Traditional formulation of
state estimation (TSE) in which the network is modeled by the bus-branch model; the
Generalized State Estimation (GSE), where the bus section level or substation level of
the network is also taken into account. In addition to those, the chapter also discusses
and portrays two different approaches for the type of measurement processed by the
state estimator. In the first instance, both TSE and GSE are formulated considering
that the set of available measurements is composed only of conventional
measurements, which are: power flow, power injection, and voltage magnitude
measurements. In the second instance, the modifications required to include
synchronized phasor measurements units (PMU) are deliberated and described.
2.1 TRADITIONAL STATE ESTIMATION
This section presents the formulation of Traditional State Estimation in which
conventional measurements, such as power flow, power injections, and voltage
magnitudes, are pondered.
2.1.1 SE with Conventional Measurements
Traditional State Estimation (TSE) refers to a procedure for obtaining all the
voltage phasors of a given power network (ABUR; EXPÓSITO, 2004). The system is
modeled by the bus-branch model; buses represent the substations, and PI models
indicate the transmission lines and transformers. Figure 1 portrays a power system at
the bus-branch model.
24
Figure 1 – IEEE 14 Bus System – Bus-branch Model Source: University of Washington (2015)
In the TSE, SCADA system gathers all the real-time measurements spread all
over the power system, and it process them in the control center. SE algorithms will
provide the most likely estimated states of the entire network if the system is
observable. Since real time measurements contain errors, and such errors have a
Gaussian (Normal) distribution, its variance depends on the measuring device
precision. Thus, the procedure for obtaining the estimated states uses a statistical
approach.
The classical and established SE method is the Weighted Least Squares
(WLS), which relies on the following measurement model, shown in Eq. (1).
𝑧 = ℎ(𝑥) + 𝑒 (1)
where:
𝑧: is the measurement vector, with size 𝑚;
𝑥: is the state vector, with size 𝑛;
ℎ(𝑥): is the nonlinear function relating the measurements to the system states;
25
𝑒: is the vector of measurements errors;
𝑚: is the number of measurements;
𝑛: is the number of states.
The state vector 𝑥 has a dimension of 2𝑁 − 1, where 𝑁 is the number of the
buses, and is given by Eq. (2).
𝑥 = [ 𝜃2 𝜃3 … 𝜃𝑛 𝑉1 𝑉2 𝑉3 … 𝑉𝑛]𝑇 . (2)
In Eq. (2) the bus 1 is chosen as the slack bus, with the phase angle set to
zero, what provides a system reference. Eq. (3) represents the measurement function
ℎ(𝑥) vector, which relates each measurement to the state variables:
ℎ(𝑥) =
[ 𝑃𝑘𝑚
𝑃𝑘
𝑄𝑘𝑚
𝑄𝑘
𝑉𝑘 ]
. (3)
where:
𝑃𝑘𝑚: refers to active power flow measurements from a generic bus 𝑘 to bus 𝑚;
𝑃𝑘: refers to active power injection measurements at a generic bus 𝑘;
𝑄𝑘𝑚: refers to reactive power flow measurements from a generic bus 𝑘 to bus 𝑚;
𝑄𝑘: refers to reactive power injection measurements at a generic bus 𝑘;
𝑉𝑘: refers to voltage magnitude measurements at a generic bus 𝑘;1.
1Please refer to Appendix A for further description of all measurement equations.
26
The purpose of the WLS SE is to obtain the state variables, which minimizes
the following objective function, presented in the Eq. (4) (ABUR; EXPÓSITO, 2004;
SCHWEPPE, FRED C; WILDES, 1970):
𝐽(𝑥) = ∑(𝑧𝑖 − ℎ𝑖(𝑥))2
𝑅𝑖𝑖⁄ =
𝑚
𝑖=1
[𝑧 − ℎ(𝑥)]𝑇𝑅−1[𝑧 − ℎ(𝑥)] (4)
where:
𝑅: is a diagonal covariance matrix, given by 𝑅 = 𝑑𝑖𝑎𝑔{𝜎12, 𝜎2
2, … , 𝜎𝑚2 }.
In others words, the WLS SE aims at minimizing the sum of weighted
measurement residues.
A minimum is found once the first-order optimality conditions are satisfied
(ABUR; EXPÓSITO, 2004), and it is achieved by a first-order approximation, given in
Eq. (5):
𝑔(𝑥) =𝜕𝐽(𝑥)
𝜕𝑥= −𝐻𝑇(𝑥)𝑅−1[𝑧𝑖 − ℎ𝑖(𝑥)] = 0 (5)
where:
𝐻(𝑥): is the Jacobian matrix, given by Eq. (6). 2
2 Please refer to Appendix A for further description of all Jacobian equations.
27
𝐻(𝑥) = 𝜕ℎ(𝑥)
𝜕𝑥⁄ =
[ 𝜕𝑃𝑘𝑚
𝜕𝜃⁄ 𝜕𝑃𝑘𝑚
𝜕𝑉⁄
𝜕𝑃𝑘𝜕𝜃
⁄ 𝜕𝑃𝑘𝜕𝑉
⁄
𝜕𝑄𝑘𝑚𝜕𝜃
⁄ 𝜕𝑄𝑘𝑚𝜕𝑉
⁄
𝜕𝑄𝑘𝜕𝜃
⁄ 𝜕𝑄𝑘𝜕𝑉
⁄
𝜕𝑉𝑘𝜕𝜃
⁄ 𝜕𝑉𝑘𝜕𝑉
⁄ ]
. (6)
By expanding the nonlinear function 𝑔(𝑥) into its Taylor series around the state
vector 𝑥𝑘, and by neglecting the higher order terms, it can lead to an iterative solution
scheme, given in Eq. (7):
𝐺(𝑥𝑘)𝛥𝑥𝑘+1 = 𝐻𝑇(𝑥𝑘)𝑊[𝑧 − ℎ(𝑥𝑘)] (7)
where:
𝐺(𝑥𝑘) = 𝐻𝑇(𝑥𝑘)𝑊𝐻(𝑥𝑘): is the Gain matrix;
𝑊 = 𝑅−1: is the weighting matrix;
𝛥𝑥𝑘 = 𝑥𝑘+1 − 𝑥𝑘, being 𝑘 the iteration index.
Abur and Expósito (2004) present a step by step algorithm to solve the
traditional state estimation problem, which is summarized as follows:
1. Set iteration index 𝑘 equal to zero;
2. Initialize the state vector 𝑥𝑘 in a flat start (voltage magnitudes equal to one and
voltage angle equal to zero);
3. Calculate the Gain matrix 𝐺(𝑥𝑘);
4. Calculate the equation 𝐻𝑇(𝑥)𝑅−1[𝑧𝑖 − ℎ𝑖(𝑥𝑘)];
5. Determine ∆𝑥𝑘;
6. Test for convergence, i.e. max|∆𝑥𝑘| ≤ 𝜖, where 𝜖 is the tolerance;
7. If the tolerance is attained, stop. If not, update 𝑥𝑘+1 = 𝑥𝑘 + ∆𝑥𝑘, 𝑘 = 𝑘 + 1, and go
back to step 3.
28
2.1.2 SE with Synchronized Phasor Measurements
The PMU advent recalls as a distance relay with symmetrical components,
developed in the 70´s at Virginia Tech Laboratory (PHADKE; IBRAHIM; HLIBKA,
1977). The capability of obtaining synchronized measurements with GPS time stamp
has developed important advances, for instance: PMU provides the magnitude and the
angle of the voltage at the bus where it is connected; it is usable as a measurement in
the state estimation equations and it does not required to set a slack bus, as reported
in (ZHU; ABUR, 2007). Hence, the state vector must include all the voltage angles and
magnitudes, with a dimension 2𝑁, as presented in Eq. (8).
𝑥 = [ 𝜃1 𝜃2 𝜃3 … 𝜃𝑛 𝑉1 𝑉2 𝑉3 … 𝑉𝑛]𝑇 . (8)
PMU not only provides the voltage phasors but it also can provide current
phasor measurements of adjacent power lines, depending upon the number of
channels (KORKALI; ABUR, 2009). This way, the current phasors may also be
included as measurements in state estimation equations.
Considering the current flow from a bus to another, one can say that it is
usually measured and transmitted in the polar form, i.e. current magnitude (𝐼𝑘𝑚) and
angle (𝛿𝑘𝑚). It is preferable, however, to use them in the rectangular form, due to
numerical problems in case of either lightly load systems or flat start initialization
(KORRES; MANOUSAKIS, 2011). Notwithstanding, the rectangular coordinates
present disadvantage as it will amplify errors of PMU measurements (KORRES;
MANOUSAKIS, 2011). Such conversion is simple to apply, as it can be seen at the set
of Eq. (9).
𝐼𝑘𝑚𝑅𝑒 = 𝐼𝑘𝑚cos (𝛿𝑘𝑚)
𝐼𝑘𝑚𝐼𝑚 = 𝐼𝑘𝑚sin (𝛿𝑘𝑚)
(9)
29
where:
𝐼𝑘𝑚𝑅𝑒 and 𝐼𝑘𝑚
𝐼𝑚 : refers to the real and imaginary parts of phasor current measurement;
𝐼𝑘𝑚: refers to the phasor current magnitude, measured by PMU;
𝛿𝑘𝑚: refers to the phasor current angle, measured by PMU.
Thus, the measurement function equations and Jacobian matrix elements
must be properly adapted to accommodate the new measurements, as follows:
ℎ(𝑥) =
[ 𝑃𝑘𝑚
𝑃𝑘
𝑄𝑘𝑚
𝑄𝑘
𝑉𝑘
𝜃𝑘
𝐼𝑘𝑚𝑅𝑒
𝐼𝑘𝑚𝐼𝑚 ]
(10)
𝐻(𝑥) = 𝜕ℎ(𝑥)
𝜕𝑥⁄ =
[ 𝜕𝑃𝑘𝑚
𝜕𝜃⁄ 𝜕𝑃𝑘𝑚
𝜕𝑉⁄
𝜕𝑃𝑘𝜕𝜃
⁄ 𝜕𝑃𝑘𝜕𝑉
⁄
𝜕𝑄𝑘𝑚𝜕𝜃
⁄ 𝜕𝑄𝑘𝑚𝜕𝑉
⁄
𝜕𝑄𝑘𝜕𝜃
⁄ 𝜕𝑄𝑘𝜕𝑉
⁄
𝜕𝑉𝑘𝜕𝜃
⁄ 𝜕𝑉𝑘𝜕𝑉
⁄
𝜕𝜃𝑘𝜕𝜃
⁄ 𝜕𝜃𝑘𝜕𝑉
⁄
𝜕𝐼𝑘𝑚𝑅𝑒
𝜕𝜃⁄ 𝜕𝐼𝑘𝑚
𝑅𝑒
𝜕𝑉⁄
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝜃⁄ 𝜕𝐼𝑘𝑚
𝐼𝑚
𝜕𝑉⁄ ]
. (11)
By deriving the current phasor flow equations as a function of the power flows,
it is possible to determine the partial derivatives of the Jacobian matrix, as follows:
30
𝐼𝑘𝑚𝑅𝑒 =
𝑃𝑘𝑚 cos(𝜃𝑘) + 𝑄𝑘𝑚sin (𝜃𝑘)
𝑉𝑘
𝐼𝑓𝑙𝑜𝑤𝐼𝑚 =
𝑃𝑘𝑚 sin(𝜃𝑘) − 𝑄𝑘𝑚cos (𝜃𝑘)
𝑉𝑘
(12)
where:
𝑉𝑘: refers to the voltage magnitude at the sending bus 𝑘, measured by the PMU;
𝜃𝑘: refers to the voltage angle at the sending bus 𝑘, measured by the PMU.
Having the above mentioned in mind, the partial derivatives of real part of the
current phasor will be:
𝜕𝐼𝑘𝑚𝑅𝑒
𝜕𝜃𝑘=
1
𝑉𝑘[cos(𝜃𝑘)(
𝜕𝑃𝑘𝑚
𝜕𝜃𝑘+ 𝑄𝑘𝑚) − sin(𝜃𝑘)(
𝜕𝑄𝑘𝑚
𝜕𝜃𝑘− 𝑃𝑘𝑚)]
𝜕𝐼𝑘𝑚𝑅𝑒
𝜕𝜃𝑚=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝜃𝑚cos(𝜃𝑘) +
𝜕𝑄𝑘𝑚
𝜕𝜃𝑚sin(𝜃𝑘)]
𝜕𝐼𝑘𝑚𝑅𝑒
𝜕𝑉𝑘=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝑉𝑘cos(𝜃𝑘) +
𝜕𝑄𝑘𝑚
𝜕𝑉𝑘sin(𝜃𝑘)] −
1
𝑉𝑘2[𝑃𝑘𝑚 cos(𝜃𝑘) + 𝑄𝑘𝑚 sin(𝜃𝑘)]
𝜕𝐼𝑘𝑚𝑅𝑒
𝜕𝑉𝑚=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝑉𝑚cos(𝜃𝑘) +
𝜕𝑄𝑘𝑚
𝜕𝜃𝑉𝑚sin(𝜃𝑘)].
(13)
Considering partial derivatives of imaginary part of the current phasor:
31
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝜃𝑘=
1
𝑉𝑘[cos(𝜃𝑘)(𝑃𝑘𝑚 −
𝜕𝑄𝑘𝑚
𝜕𝜃𝑘) + sin(𝜃𝑘)(𝑄𝑘𝑚 +
𝜕𝑃𝑘𝑚
𝜕𝜃𝑘)]
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝜃𝑚=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝜃𝑚sin(𝜃𝑘) −
𝜕𝑄𝑘𝑚
𝜕𝜃𝑚cos(𝜃𝑘)]
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝑉𝑘=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝑉𝑘sin(𝜃𝑘) −
𝜕𝑄𝑘𝑚
𝜕𝑉𝑘cos(𝜃𝑘)] −
1
𝑉𝑘2[𝑃𝑘𝑚 sin(𝜃𝑘) − 𝑄𝑘𝑚 cos(𝜃𝑘)]
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝑉𝑚=
1
𝑉𝑘[𝜕𝑃𝑘𝑚
𝜕𝑉𝑚sin(𝜃𝑘) −
𝜕𝑄𝑘𝑚
𝜕𝜃𝑉𝑚cos(𝜃𝑘)].
(14)
Moreover, there are the partial derivatives of voltage angle measurements,
which are linearly related to the states, so that:
𝜕𝜃𝑘
𝜕𝜃𝑘= 1,
𝜕𝜃𝑘
𝜕𝜃𝑚= 0,
𝜕𝜃𝑘
𝜕𝑉𝑘= 0,
𝜕𝜃𝑘
𝜕𝑉𝑚= 0 (15)
After such modifications, the process for obtaining the states is the same as
that one from the previous section, applying the normal equation and performing the
previous algorithm.
2.2 GENERALIZED STATE ESTIMATION
Basically, there are three steps for real time modeling of a power network: (i)
network configuration analysis; (ii) observability analysis; (iii) state estimation and bad
data processing (MONTICELLI, 1993a). In the first one, a topology processor gathers
all the logical information from switches and circuit breakers (CB) status, so that it
forms the bus-branch model and it performs the next steps. In spite of it, the topology
processor may create an incorrect network model if a wrong status of a CB arises,
hampering all results obtained with such a model.
The generalized approach has been developed to circumvent topology
problems that cannot be detected by the topology processor. As proposed by Monticelli
and Garcia (1991), the switches and circuit breakers are modeled in conjunction with
PI models of transmission lines and transformers, in the so-called bus-
32
section/switching-device level, or substation level model. Figure 2 shows the IEEE 14
bus system at the substation level.
Figure 2 – IEEE 14 Bus System at the Substation Level Source: (CARO; CONEJO; ABUR, 2010)
Furthermore, power flows through switches and circuit breakers (from now on
referred to as switching branches) are treated as state variables to be estimated within
such approach. Thus, the use of infinite and null values of impedances in the model
might be disregarded, since they cause numerical ill-conditioning problems, as pointed
out in Monticelli and Garcia (1991). Figure 3 shows a switching branch between buses
k and l. In this case, the active and reactive power flow from bus k to l are included as
new state variables.
33
Figure 3 – Closed Switch/Breaker from Bus 𝑘 to 𝑙
By doing so, the use of branches with atypical values is avoided, while the size
of the state vector enlarges, as shown in Eq. (16).
𝑥 = [𝜃2 𝜃3 … 𝜃𝑛 𝑉1 𝑉2 𝑉3 … 𝑉𝑛 … 𝑡𝑘𝑙 𝑢𝑘𝑙]𝑇 (16)
where:
𝑡𝑘𝑙: indicates active power flow through switching branch 𝑘 − 𝑙;
𝑢𝑘𝑙: indicates reactive power flow through switching branch 𝑘 − 𝑙.
So far it has been discussed the inclusion of switches and circuit breakers in
the network model for state estimation purposes. The following subsections depict two
different formulations of the GSE: one that uses conventional measurements, and
another one that combines the use of conventional and synchronized phasor
measurements.
2.2.1 GSE with Conventional Measurements
Taken into account only conventional measurements, the available power flow
measurements on switching branches are no longer modeled as a function of voltage
phasors. Instead, they are directly related to the new state variables. Referring to
Figure 4, Eq. (17) and (18) demonstrate the GSE approach when it comes to power
flow measurement through a switching branch.
34
Figure 4 – Power flow measurements on a switching branch
𝑧𝑡𝑘𝑙= 𝑡𝑘𝑙 + 휀𝑡𝑘𝑙
(17)
𝑧𝑢𝑘𝑙= 𝑢𝑘𝑙 + 휀𝑢𝑘𝑙
(18)
Injection measurements on boundary buses, i.e. buses connecting switching
branches with transmission lines, must consider the sum of power flows in conventional
branches and the power flows through switching branches. Referring to Figure 5, Eq.
(19) and (20) demonstrate the formulation. The ticker line is a conventional branch,
with a PI model.
Figure 5 – Power Injection Measurement in a boundary bus
𝑧𝑃𝑘= ∑ 𝑃𝑘𝑚(𝜃𝑘, 𝜃𝑚, 𝑉𝑘, 𝑉𝑚)
𝑚∈𝛺𝑘
+ ∑ 𝑡𝑘𝑙 +
𝑙𝜖𝛤𝑘
휀𝑃𝑘 (19)
35
𝑧𝑄𝑘= ∑ 𝑄𝑘𝑚(𝜃𝑘 , 𝜃𝑚, 𝑉𝑘, 𝑉𝑘𝑚)
𝑚∈𝛺𝑘
+ ∑ 𝑢𝑘𝑙 +
𝑙𝜖𝛤𝑘
휀𝑄𝑘 (20)
where:
𝑃𝑘𝑚 and 𝑄𝑘𝑚: are power flows through conventional branch 𝑘 − 𝑚;
𝑡𝑘𝑙 and 𝑢𝑘𝑙: are power flows through switching branch 𝑘 − 𝑙;
𝛺𝑘: is the set of conventional branches incident to bus 𝑘;
𝛤𝑘: is the set of switch/breaker branches incident to bus 𝑘.
The status of CB can be modeled as pseudo measurements (MONTICELLI,
1993b) with high weights, or as operational constraints in an optimization problem
(SIMÕES COSTA; LOURENÇO; CLEMENTS, 2002).
If a CB is closed (Figure 3), the angle difference and voltage drop between its
nodes are set equal to zero, as presented in the set of Eq. (21). On the other hand, if
the CB is open (Figure 6), the power flows through it is set equal to zero, as in the set
of Eq. (22).
𝛥𝜃𝑘𝑙𝑝 = 𝜃𝑘 − 𝜃𝑙 = 0
𝛥𝑉𝑘𝑙𝑝 = 𝑉𝑘 − 𝑉𝑙 = 0
(21)
Figure 6 - Open Switch/Breaker from Bus 𝑘 to
36
𝑡𝑘𝑙𝑝 = 0 𝑢𝑘𝑙
𝑝 = 0 (22)
If the status of a CB is unknown, the pseudo measurements of the set of Eq.
(21) and (22) cannot be used, and the GSE must calculate the flow through it in order
to determine its status.
Moreover, the network configuration also allows the use of pseudo injection
measurements in nodes inside the substation or on the boundary, which are very
abundant when considering the approach at the substation level. In this way, the
injection in those nodes is set equal to zero, as presented in the set of Eq. (23), or it
can also be modeled as structural constraints in an optimization problem (SIMÕES
COSTA; LOURENÇO; CLEMENTS, 2002).
𝑃𝑘𝑝 = 0 𝑄𝑘
𝑝 = 0 (23)
In such case, the measurement function vector and the Jacobian matrix also
change, as they reflect the use of the new states and measurements, as shown in. Eq.
(24) and Eq. (25), respectively.
ℎ(𝑥) =
[ 𝑃𝑘𝑚
𝑃𝑘
𝑄𝑘𝑚
𝑄𝑘
𝑉𝑘
𝑡𝑘𝑙
𝑢𝑘𝑙
𝑃𝑘𝑝
𝑄𝑘𝑝
𝛥𝜃𝑘𝑙𝑝
𝛥𝑉𝑘𝑙𝑝
𝑡𝑘𝑙𝑝
𝑢𝑘𝑙𝑝
]
(24)
37
𝐻 =
[ 𝜕𝑃𝑘𝑚
𝜕𝜃⁄ 𝜕𝑃𝑘𝑚
𝜕𝑉⁄ 𝜕𝑃𝑘𝑚
𝜕𝑡𝑘𝑙⁄
𝜕𝑃𝑘𝑚𝜕𝑢𝑘𝑙
⁄
𝜕𝑃𝑘𝜕𝜃
⁄ 𝜕𝑃𝑘𝜕𝑉
⁄ 𝜕𝑃𝑘𝜕𝑡𝑘𝑙
⁄𝜕𝑃𝑘
𝜕𝑢𝑘𝑙⁄
𝜕𝑄𝑘𝑚𝜕𝜃
⁄ 𝜕𝑄𝑘𝑚𝜕𝑉
⁄ 𝜕𝑄𝑘𝑚𝜕𝑡𝑘𝑙
⁄𝜕𝑄𝑘𝑚
𝜕𝑢𝑘𝑙⁄
𝜕𝑄𝑘𝜕𝜃
⁄ 𝜕𝑄𝑘𝜕𝑉
⁄ 𝜕𝑄𝑘𝜕𝑡𝑘𝑙
⁄𝜕𝑄𝑘
𝜕𝑢𝑘𝑙⁄
𝜕𝑉𝑘𝜕𝜃
⁄ 𝜕𝑉𝑘𝜕𝑉
⁄ 𝜕𝑉𝑘𝜕𝑡𝑘𝑙
⁄𝜕𝑉𝑘
𝜕𝑢𝑘𝑙⁄
𝜕𝑡𝑘𝑙𝜕𝜃
⁄ 𝜕𝑡𝑘𝑙𝜕𝑉
⁄ 𝜕𝑡𝑘𝑙𝜕𝑡𝑘𝑙
⁄𝜕𝑡𝑘𝑙
𝜕𝑢𝑘𝑙⁄
𝜕𝑢𝑘𝑙𝜕𝜃
⁄ 𝜕𝑢𝑘𝑙𝜕𝑉
⁄ 𝜕𝑢𝑘𝑙𝜕𝑡𝑘𝑙
⁄𝜕𝑢𝑘𝑙
𝜕𝑢𝑘𝑙⁄
𝜕𝑃𝑘𝑝
𝜕𝜃⁄ 𝜕𝑃𝑘
𝑝
𝜕𝑉⁄ 𝜕𝑃𝑘
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝑃𝑘𝑝
𝜕𝑢𝑘𝑙⁄
𝜕𝑄𝑘𝑝
𝜕𝜃⁄ 𝜕𝑄𝑘
𝑝
𝜕𝑉⁄ 𝜕𝑄𝑘
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝑄𝑘𝑝
𝜕𝑢𝑘𝑙⁄
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝜃⁄ 𝜕𝛥𝜃𝑘𝑙
𝑝
𝜕𝑉⁄ 𝜕𝛥𝜃𝑘𝑙
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑢𝑘𝑙⁄
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝜃⁄ 𝜕𝛥𝑉𝑘𝑙
𝑝
𝜕𝑉⁄ 𝜕𝛥𝑉𝑘𝑙
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑢𝑘𝑙⁄
𝜕𝑡𝑘𝑙𝑝
𝜕𝜃⁄ 𝜕𝑡𝑘𝑙
𝑝
𝜕𝑉⁄ 𝜕𝑡𝑘𝑙
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝑡𝑘𝑙𝑝
𝜕𝑢𝑘𝑙⁄
𝜕𝑢𝑘𝑙𝑝
𝜕𝜃⁄ 𝜕𝑢𝑘𝑙
𝑝
𝜕𝑉⁄ 𝜕𝑢𝑘𝑙
𝑝
𝜕𝑡𝑘𝑙⁄
𝜕𝑢𝑘𝑙𝑝
𝜕𝑢𝑘𝑙⁄
]
. (25)
The set of the Eq. (26) present some of the derivatives3:
3 The same equations apply for partial derivatives of pseudo power flows measurements.
38
𝜕𝑡𝑘𝑙
𝜕𝜃𝑘=
𝜕𝑡𝑘𝑙
𝜕𝜃𝑚=
𝜕𝑡𝑘𝑙
𝜕𝜃𝑙=
𝜕𝑡𝑘𝑙
𝜕𝑉𝑘=
𝜕𝑡𝑘𝑙
𝜕𝑉𝑚=
𝜕𝑡𝑘𝑙
𝜕𝑉𝑙=
𝜕𝑡𝑘𝑙
𝜕𝑢𝑘𝑙= 0,
𝜕𝑡𝑘𝑙
𝜕𝑡𝑘𝑙= 1
𝜕𝑢𝑘𝑙
𝜕𝜃𝑘=
𝜕𝑢𝑘𝑙
𝜕𝜃𝑚=
𝜕𝑢𝑘𝑙
𝜕𝜃𝑙=
𝜕𝑢𝑘𝑙
𝜕𝑉𝑘=
𝜕𝑢𝑘𝑙
𝜕𝑉𝑚=
𝜕𝑢𝑘𝑙
𝜕𝑉𝑙=
𝜕𝑢𝑘𝑙
𝜕𝑡𝑘𝑙= 0,
𝜕𝑢𝑘𝑙
𝜕𝑢𝑘𝑙= 1
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝜃𝑚=
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑉𝑘=
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑉𝑚=
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑉𝑙=
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑡𝑘𝑙=
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝑢𝑘𝑙= 0,
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝜃𝑘= 1,
𝜕𝛥𝜃𝑘𝑙𝑝
𝜕𝜃𝑙
= −1
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝜃𝑘=
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝜃𝑚=
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝜃𝑙=
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑉𝑚=
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑡𝑘𝑙=
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑢𝑘𝑙= 0,
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑉𝑘= 1,
𝜕𝛥𝑉𝑘𝑙𝑝
𝜕𝑉𝑙
= −1.
(26)
To estimate the states, the same equation showed in (7) is iteratively solved
following the same steps presented in subsection 2.1.1.
GSE approach is similar to the conventional WLS SE, except for the fact that
the network model contains switches and circuit breakers, and the power flows through
switching branches are state variables. By doing so, the number of states enlarges and
so does the size of Jacobian matrix and the measurements function; such changing
takes place due to the use of pseudo measurements to represent switch status and
null injection nodes (MONTICELLI, 1993a).
The GSE based on the normal equation formulation uses angle difference and
voltage drop across zero impedance branches (closed CB), zero power flows across
infinite branches (opened CB), and zero injection measurements with high weighting
factors, in order to attain acceptable accuracy (MONTICELLI; GARCIA, 1991).
Another approach suggests the use of such pseudo measurements as equality
constraints for an optimization problem, as presented in Eq. (27).
39
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 1
2𝑟𝑇𝑊𝑟
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑟 = 𝑧 − ℎ(�̂�)
ℎ𝑜(�̂�) = 0
ℎ𝑠(�̂�) = 0
(27)
where:
𝑥: is the vector of estimated states;
ℎ𝑜: is the vector of operational constraints;
ℎ𝑠: is the vector of structural constraints.
Operational constraints stand for the status of CB (angle difference, voltage
drop, and power flows), and structural constraints stand for null injection
measurements and reference bus. The Hacthel´s sparse tableau algorithm
(CLEMENTS; SIMÕES COSTA, 1998; GJELSVIK; HOLTEN, 1985) solves this
constrained nonlinear problem.
2.2.2 GSE with Phasor Measurements
When voltage and/or current synchronized phasor measurements are
available inside the substations, the WLS problem formulated in the subsection 2.2.1
must be adapted. The equations derived ahead consist of one of the contributions of
this work, since they are not easily found in papers and books.
Bearing in mind the measurement arrangement in the substation represented
in Figure 7, it can be seen that the voltage phasor measurement at busbar 2 provides
a voltage magnitude measurement, as well as a phase angle at the same bus. Its
implementation is simple and straightforward, as it only needs a voltage angle as a
measurement.
The current phasor measurements provide the current magnitude and angle
of the current flowing from a busbar to another. As suggested before, these current
phasor measurements in the polar form can be converted into a rectangular form (see
40
Eq. (9)). Thus, two formulations for the WLS SE, complying with those measurements,
are offered as follows.
Figure 7 – Substation model with phasor and conventional measurements
(a) Considering power flows through the switching branches as state variables
In such case, the active and reactive power flows through the switching
branches are kept as state variables. However, the available current phasor
measurements on switching branches 2-4 and 2-5 are no longer power flows, but
current flows instead. Then, its corresponding partial derivatives related to power flows
are not linear and must be derived as a function of them.
It is preferable to represent the current phasor measurements as a function of
the state variables to derive its partial derivatives, as follows.
41
𝐼𝑘𝑙𝑅𝑒 =
𝑡𝑘𝑙 cos(θ𝑘) + 𝑢𝑘𝑙sin (θ𝑘)
𝑉𝑘
𝐼𝑘𝑙𝐼𝑚 =
𝑡𝑘𝑙 sin(θ𝑘) − 𝑢𝑘𝑙cos (θ𝑘)
𝑉𝑘
(28)
where:
𝑡𝑘𝑙: refers to the active power flows through the switching branch (state variable);
𝑢𝑘𝑙: refers to the reactive power flows through the switching branch (state variable);
𝑉𝑘: refers to the voltage magnitude at the measured bus, by the PMU;
θ𝑘: refers to the voltage angle at the measured bus, by the PMU.
Then, the partial derivatives are easily obtained:
𝜕𝐼𝑘𝑙𝑅𝑒
𝜕𝑡𝑘𝑙=
cos(𝜃𝑘)
𝑉𝑘
𝜕𝐼𝑘𝑙𝑅𝑒
𝜕𝑢𝑘𝑙=
sin(𝜃𝑘)
𝑉𝑘
𝜕𝐼𝑘𝑙𝐼𝑚
𝜕𝑡𝑘𝑙=
sin(𝜃𝑘)
𝑉𝑘
𝜕𝐼𝑘𝑚𝐼𝑚
𝜕𝑢𝑘𝑙= −
cos(𝜃𝑘)
𝑉𝑘
(29)
The power injection measurement at the busbar 3, and the power flow
measurements from the same busbar for nodes 4 and 5, are linearly represented as a
function of the state variables.
The Jacobian matrix of the system, presented in Figure 7, takes the form
present in Eq. (30) and only the active part is shown, for the sake of simplicity.
42
It is worth mentioning that in such approach, there is no need to extract a
column to provide a reference for the system, once there is at least one synchronized
voltage phasor measurement (ZHU; ABUR, 2007).
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝐻 =
𝑃1−4
𝑃3−4
𝑃5−3
𝑃3
𝐼1−5𝑅𝑒
𝐼2−4𝑅𝑒
𝐼2−5𝑅𝑒
𝜃1
𝜃2
𝑃4𝑝
𝑃5𝑝
∆𝜃2−4𝑝
∆𝜃3−5𝑝
𝑡3−4𝑝
𝑡2−5𝑝 [
∗ ∗
1
−1
1 1
∗ ∗
∗
∗
1
1
∗ ∗ −1 −1
∗ ∗ −1 −1
1 −1
1 −1
1
1 ]
.
(30)
where: * refers to non-zero elements
(b) Considering current flows through the switching branches as state variables
In such alternative, the real and imaginary parts of the currents through
switching branches are used as state variables, instead of the active and reactive
power flows. Since more phasor measurements will be available in the future, it is
assumed that an entire substation is measured only by such devices. For that reason,
it is reasonable to make the proposed changes in the formulation of the generalized
state estimation, as suggested in Yang, Sun and Bose (2011).
In this case, the current phasor measurements on switching branches are
linearly related to the state variables, though the power flow measurements need to be
derived as a function of the new state variables.
43
Equation (31) represents the power flows as a function of the current and the
voltage:
𝑃𝑘𝑙 = 𝑉𝑘[𝐼𝑘𝑙𝑅𝑒 cos(θ𝑘) + 𝐼𝑘𝑙
𝐼𝑚sin (θ𝑘)]
𝑄𝑘𝑙 = 𝑉𝑘[𝐼𝑘𝑙𝑅𝑒 sin(θ𝑘) − 𝐼𝑘𝑙
𝐼𝑚cos (θ𝑘)].
(31)
The derivatives are taken straightforward, as follows:
𝜕𝑃𝑘𝑙
𝜕𝐼𝑘𝑙𝑅𝑒
= 𝑉𝑘 cos(𝜃𝑘)
𝜕𝑃𝑘𝑙
𝜕𝐼𝑘𝑙𝐼𝑚
= 𝑉𝑘 sin(𝜃𝑘)
𝜕𝑄𝑘𝑙
𝜕𝐼𝑘𝑙𝑅𝑒
= 𝑉𝑘 sin(𝜃𝑘)
𝜕𝑄𝑘𝑙
𝜕𝐼𝑘𝑙𝐼𝑚
= −𝑉𝑘 cos(𝜃𝑘)
(32)
By making use of the same system of Figure 7, the Jacobian matrix takes on
the form of Equation (33).
44
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝐼24 𝐼25 𝐼34 𝐼35
𝐻 =
𝑃1−4
𝑃3−4
𝑃5−3
𝑃3
𝐼1−5𝑅𝑒
𝐼2−4𝑅𝑒
𝐼2−5𝑅𝑒
𝜃1
𝜃2
𝑃4𝑝
𝑃5𝑝
∆𝜃2−4𝑝
∆𝜃3−5𝑝
𝐼3−4𝑅𝑒,𝑝
𝐼2−5𝑅𝑒,𝑝 [
∗ ∗
∗
∗
∗ ∗
∗ ∗
1
1
1
1
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
1 −1
1 −1
1
1 ]
.
(33)
As it can be noticed, on one hand, the phasor current flow measurements
through switching branches 2-4 and 2-5 are linearly related to the state variables. On
the other hand, the power flow measurements through switching branches 3-4 and 3-
5 must be derivative as a function of the currents, accordingly to what had been set on
Eq. (32).
In its turn, the power injection measurements at boundary buses/nodes and
busbars connecting only switching branches are no longer linearly related to the state
variables. They are derived as a function of the current flows of the adjacent switches.
The power injection measurement at busbar 3 is formulated as a summation of
the flows through the switching branches 3-4 and 3-5. In the previous approach, they
were linearly related to the state variables, but for the sake of the current approach,
those power flows must be derived as a function of the current flows, which are the
new state variables. The expression for these derivatives is the same from that of the
power flows, as presented at the set of Eq.(32).
In the case of pseudo injection measurement at the boundary bus/node 4 and
5, both can be represented as a summation of the power flows through the switching
branches and conventional branches. Thus, its derivatives follow the same approach,
45
with elements referring to conventional state variables (voltage angle and magnitude)
and the new ones (current flows through switching branches).
Another difference that can be pinpointed is that the pseudo flow
measurements through the opened CB can be directly used as pseudo current flow
measurements, once there are set equal to zero.
2.3 SUMMARY OF THE CHAPTER
This chapter has exposed the theory around the State Estimation paradigm
when it comes to the Traditional and Generalized approaches. The Traditional
approach was first presented in order to demonstrate the classical SE over the WLS
method. It was also unfolded the extension of complying with the phasor
measurements, which have been benefiting the SE algorithms.
In the sequence, the Generalized approach was discussed, considering both
conventional and phasor measurements. It had been illustrated how switching
branches can be modeled to perform the state estimation, pondering the new state
variables, and the new pseudo measurements, operational and structural constraints.
Moreover, the use of phasor measurements was presented considering two different
approaches: 1) by changing the state variables from power flows through switching
branches to current flows, and 2) by converting the power injection measurement into
current injections.
The equations derived here are used in the developed GSE algorithm, which
validates the methods proposed on the subsequent chapters.
46
3 OBSERVABILITY ANALYSIS
The effectiveness of performing state estimation on a power network depends
on the availability of enough and well distributed measurements throughout the system
(MONTICELLI, 1999). Conventionally, the observability analysis is carried out prior to
the state estimation execution, and once the system is observable, it enables to
perform further analysis. However, if the system is unobservable, it is yet useful to
determine what portions are observable, as well as which parts are unobservable. As
a consequence, it is possible to perform a partial state estimation, or use
pseudomeasurements to restate the system observability.
This chapter addresses the concepts of Observability analysis in power
networks modeled at the bus-branch and bus section levels. It discusses the numerical
and topological approaches to determine the system observability along with further
numerical methods to find unobservable branches and observable islands. In addition
to that, observability methods are explained in tutorial examples, since they are the
basis of this work developments.
3.1 TRADITIONAL OBSERVABILITY ANALYSIS: BUS-BRANCH ANALYSIS
In real time modeling, topology processor reduces the network from the bus
section level into the bus-branch model, as it processess the status of switches and
circuit breakers inside the substations. After that, the Traditional State Estimation is
performed.
One of the first papers addressing observability issues calls for a topological
approach by making usage of an iterative algorithm whose aim is to find a spanning
tree of full rank (KRUMPHOLZ; CLEMENTS; DAVIS, 1980). Conversely, the approach
proposed by Monticelli and Wu (1985a, 1985b) claims for a numerical approach to
determine the system observability by applying the triangular factorization in the Gain
matrix.
47
3.1.1 Network Observability
For observability purposes, it is convenient to use a simplified linearized model
that represents only the active part, and assuming 1.0 p.u. reactances (MONTICELLI;
WU, 1985a). The reactive part should also be tested, but since the measurements
usually come in pairs, the second part is seldom necessary (MONTICELLI; WU,
1985b).
When it comes to Traditional Observability Analysis (TOA), the first step to be
taken is the modeling of the measurements. There are three type of measurements: (i)
analog, consisted by power flows, power injections, bus voltage magnitude, current
magnitude, and also synchronized voltage and current phasor measurements; (ii)
logical, composed by the status of switches and circuit breakers and (iii)
pseudomeasurements, consisted by forecasted bus loads and generations
(MONTICELLI; WU, 1985a).
Logical information is used for topology processing and it is performed prior
the observability analysis. Pseudo measurements become relevant once the system
is found unobservable, and are used to restore observability. Therefore, only the first
type of measurements is considered for observability analysis and their modeling is
addressed as follows (MONTICELLI; WU, 1985a)4.
(a) Power flow measurements
Figure 8 presents a line model connecting buses 𝑘-𝑚, with a power flow
measurement on it.
4 In this chapter, only the conventional measurements are presented. For PMU measurements, see next chapter.
48
Figure 8 – Power flow measurement on branch 𝑘 − 𝑚
Assuming the line reactance (𝑥𝑘−𝑚) equal to 1.0 p.u., the following equation
represents the linear power flow measurement.
𝑃𝑘−𝑚𝑚 =
1
𝑥𝑘−𝑚
(𝜃𝑘 − 𝜃𝑚) = 𝜃𝑘 − 𝜃𝑚 (34)
(b) Power injection measurements
A power injection measurement at a bus is modeled as a summation of all
power flows from all adjacent lines. Taking for instance the power injection
measurement set at the generic bus 𝑡 of the 3 bus system in Figure 9, its linear model
is given by Eq. (35).
Figure 9 – Power injection measurement at the generic bus 𝑡
49
𝑃𝑡𝑚 = ∑𝑃𝑖 =
𝑛𝑡
𝑖
1
𝑥𝑡−𝑘
(𝜃𝑡 − 𝜃𝑘) +1
𝑥𝑡−𝑚
(𝜃𝑡 − 𝜃𝑚) = 2𝜃𝑡 − 𝜃𝑘 − 𝜃𝑚 (35)
where:
𝑛𝑡: refers to a set of all branches connected to bus 𝑡.
The measurements can be grouped in a matrix form to facilitate numerical
analysis, such as the one presented ahead. Taking the system in Figure 10 as an
example, with the given measurement design, the measurement matrix is formed as
follows.
Figure 10 – 3 bus network
50
3.1.1.1 – Basic Numerical Method
Eq. (36) represents the Jacobian matrix (measurement matrix) of the given
measurement design. 5
𝜃1 𝜃2 𝜃3
𝐻𝐴𝐴 =𝑃1−2
𝑃1−3
𝑃3
[1 −11 −1
−1 −1 2
]. (36)
The reactive model requires an additional measurement, that is, the voltage
magnitude. The voltage angle, in its turn, corresponds to the same in the active model.
Computing the rank of the Jacobian matrix in Eq. (36) it is possible to
determine whether the network is observable or not, regarding the available
measurements. If the Jacobian matrix has a full rank (i.e. 𝑟𝑎𝑛𝑘(𝐻𝐴𝐴) = 𝑛 = 𝑁𝑏 − 1,
where 𝑛 is the number of states and 𝑁𝑏 the number of buses) the system is said
observable. In the example of Figure 10, the corresponding rank is full (𝑟𝑎𝑛𝑘(𝐻𝐴𝐴) =
𝑁𝑏 − 1 = 2), rendering the system observable.
Another way to determine the system observability is by computing the Gain
matrix and performing the triangular factorization, as in Eq. (37) and (38)
(MONTICELLI; WU, 1985b).
𝜃1 𝜃2 𝜃3
𝐺 = 𝐻𝐴𝐴𝑇𝐻𝐴𝐴 =
𝜃1
𝜃2
𝜃3
[3 −3
2 −2−3 −2 5
] (37)
5The Jacobian matrix 𝐻𝐴𝐴 is the same defined in Eq (6), but in a linearized fashion.
51
𝜃1 𝜃2 𝜃3
𝑈 =
𝜃1
𝜃2
𝜃3
[1.7 −1,7
1.4 −1.40
]. (38)
The existence of only one zero pivots, i.e. a zero in the diagonal of 𝑈 matrix,
renders the system observable. Such fact indicates that only one angular reference is
required, that is, an angle constraint (𝜃3 = 0 for instance), and it provides an angular
reference. Having more zero pivots, the system is unobservable (MONTICELLI; WU,
1985a).
3.1.1.2 – Basic Topological Method
In the topological approach, the power flow and injection measurements are
processed as edges connecting the vertices (representing network buses) so as to
form an observable spanning tree (KRUMPHOLZ; CLEMENTS; DAVIS, 1980).
Basically, a power flow measurement between buses 𝑘 and 𝑚, such as presented in
Figure 8, is processed as an edge connecting the related vertices 𝑘 and 𝑚, as shown
in Figure 11.
Figure 11 – Edge of a power flow measurement
Power injection measurements, on the other hand, can form edges with all the
adjacent vertices. In Figure 9, for instance, the injection measurement at bus 𝑡 forms
edges connecting vertices 𝑘 and 𝑚. However, only one edge can be used to ensure
the network observability. In Figure 12, only one of the two edges can be picked.
52
Figure 12 – Edges of a power injection measurement
Observability analysis is carried out considering the 𝑃 − 𝜃/𝑄 − 𝑉 decoupling
principle, since the measurements come in pairs. To accomplish the analysis, an
angular reference must be provided for 𝑃 − 𝜃 observability, and at least one voltage
magnitude measurement is required to ensure 𝑄 − 𝑉 observability. They are treated
as a fictitious flow, which connects a vertex to an extra (ground) node (CLEMENTS;
DAVIS; KRUMPHOLZ, 1981). Figure 13 portrays the measurement graph of the
network shown in Figure 10, which forms an Observable Spanning Tree (OST).
Figure 13 – Measurement graph of 3 bus network
53
3.1.2 Identification of Observable Islands
Further analysis can be carried out, for instance, measurement criticality, state
estimation, bad data and so forth, if a network is observable. On the other hand, if the
network is unobservable, it is desirable to find the observable islands and
unobservable branches.
The first step consists of identifying and removing the irrelevant branches with
the absence of flow measurements, and injection measurements on its adjacent buses.
Taking for instance the example of Monticelli and Wu (1985b) presented in Figure 14,
the branch 2-3 is found irrelevant and its corresponding row is eliminated from
incidence matrix 𝐴, in Eq. (39).
Figure 14 – 6 bus network example Source: Monticelli and Wu (1985b)
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝜃6
𝐴 =
𝑏1−2
𝑏1−3
𝑏3−4
𝑏4−5
𝑏6−4[ 1 −11 −1
1 −11 −1
−1 1 ]
(39)
54
In the next step, the measurements are processed forming the Jacobian matrix
𝐻𝐴𝐴 and the corresponding Gain matrix 𝐺, as presented in Eq. (40) and (41),
respectively.
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝜃6
𝐻𝐴𝐴 =
𝑃1
𝑃4
𝑃1−2
𝑃4−5
[
2 −1 −1−1 3 −1 −1
1 −11 −1
] (40)
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝜃6
𝐺 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝜃6 [
5 −3 −2−3 2 1−2 1 2 −3 1 1
−3 10 −4 −31 −4 2 11 −3 1 1 ]
. (41)
At this point, is possible to evaluate the rank of matrix 𝐻 and to certify the
system unobservability. Performing the triangular factorization of the Gain matrix 𝐺,
two zero pivots are found, as follows.
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝜃6
𝑈 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝜃6 [ 2.24 −1.34 −0.89
0.45 −0.451 −3 1 1
1 −10
0 ]
. (42)
Those two zero pivots indicate that the system is divided into two observable
islands (MONTICELLI; WU, 1985b), and two angular references must be provided, on
buses 5 and 6.
Replacing the zero pivots by “1”s, at positions (5,5) and (6,6) of 𝑈 matrix, and
changing the corresponding number in the right vector 𝑡𝐴 by random numbers, one can
determine the unobservable state, as in Eq. (43).
55
𝜃 = (𝑈𝑇𝑈)−1𝑡𝐴 (43)
where:
𝑡𝐴 = [0 0 0 0 0 1]𝑇
The vector 𝑡𝐴 is a modified vector, with zeros in the positions related to non-
zero pivots, and random numbers on the positions related to zero pivots. Integer
numbers such as 0,1,2, etc are usually used (ABUR; EXPÓSITO, 2004).
Multiplying the estimated state 𝜃 by the incidence matrix 𝐴, it obtains the power
flows through the branches, as follows.
𝑃𝑏 = 𝐴𝜃 =
𝑏1−2
𝑏1−3
𝑏3−4
𝑏4−5
𝑏6−4[ 1 −11 −1
1 −11 −1
−1 1 ]
[ 000001]
=
[
00
−101 ]
. (44)
where:
𝑏𝑘−𝑚: refers to a branch from generic bus 𝑘 to 𝑚.
The non-zero flows indicate the unobservable branches. In such case,
branches 3-4 and 6-4 are unobservable and the corresponding row are removed from
matrix 𝐴.
After that, the irrelevant injection measurements must be identified and
removed from the set of interest. The 𝐻𝐴𝐴 and 𝐴 matrices are updated by removing the
rows related to the irrelevant injection measurements in 𝐻𝐴𝐴, and unobservable
branches in 𝐴. Therefore, the power flows are computed again, unobservable branches
are removed, as well as irrelevant measurements, until no non-zero flows are found.
In the example of Figure 14, the injection measurement at bus 4 is irrelevant,
since it has at least one adjacent unobservable branch (MONTICELLI; WU, 1985b).
56
By adapting the matrix 𝐻𝐴𝐴, processing the new Gain matrix and performing the
triangular factorization, the new 𝑈 matrix is obtained, as follows:
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝜃6
𝑈 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝜃6 [ 2.24 −1.34 −0.89
0.45 −0.450
1 −10
0 ]
. (45)
Differently, three zero pivots are found and by changing them into one, and
applying the Eq. (43) with the following vector 𝑡𝐴 = [0 0 0 0 1 2], gives 𝜃 = [0 0 0 1 1 2]
so that the power flows are evaluated as follows.
𝑃𝑏 = 𝐴𝜃 =
𝑏1−2
𝑏1−3
𝑏4−5
[1 −11 −1
1 −1
]
[ 000112]
= [000]. (46)
As a consequence, the process stops as all the power flows are null.
Although the process has stopped, it is necessary to determine if the irrelevant
branches that were removed at the beginning of the process, are observable or not. In
this example, only the branch 2-3 were labeled irrelevant. Updating the 𝐴 matrix, taking
into account the irrelevant branch and removing the already labelled unobservable
branches, we have the following matrix, shown in Eq. (47):
57
𝑃𝑏 = 𝐴𝜃 =
𝑏1−2
𝑏1−3
𝑏4−5
𝑏2−3
[
1 −11 −1
1 −11 −1
]
[ 000112]
= [
0000
]. (47)
Since all the power flows are null, all those branches are labeled observable.
The islands are identified by selecting the appropriate buses from the estimated vector
𝜃. In this case, 𝜃1, 𝜃2 and 𝜃3 form an island, 𝜃4 and 𝜃5 form another, and bus 𝜃6 forms
the last one, as depicted in Figure 15.
Figure 15 - 6 bus network example, with indication of islands
As one may notice, bus 6 is isolated from the others. Such fact does not
necessarily mean that it is not an island. As pointed out in Monticelli and Wu (1985b)
even an isolated bus, such as bus 6, is also considered an island.
58
3.2 GENERALIZED OBSERVABILITY ANALYSIS
When it comes to the problem of representing some substations in the bus
section level as a means to perform the Generalized State Estimation, all the
fundamental facts of observability analysis remain faithful (MONTICELLI, 1993a). For
the sake of this analyses, however, the model must be extended, as presented in this
section.
3.2.1 Generalized Network Observability
Although there is no problem in adding power flows through switching
branches as state variables, these new states must also be observable (MONTICELLI,
1993b). Taking the network in Figure 16 as an example, it is possible to demonstrate
how the observability methods can be extended to power networks modeled at the bus
section level.
Figure 16 – Example network at bus section level
The Jacobian matrix of the network in Figure 16 is represented in Eq. (48).
Notice that only the active part is demonstrated, since it is assumed measurements
that come in pairs.
59
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝐻𝐴𝐴 =
𝑃1−4
𝑃1
𝑃2−4
𝑃2
𝑃4
𝑝
𝑃5
𝑝
∆𝜃2−4
𝑝
∆𝜃3−5
𝑝
𝑡3−4
𝑝
𝑡2−5
𝑝 [
1 −1
2 −1 −1
1
1 1
−1 1 −1 −1
−1 1 −1 −1
1 −1
1 −1
1
1 ]
. (48)
Evaluating the rank of the matrix (48), one realizes that a full rank is achieved
(i.e. 𝑟𝑎𝑛𝑘(𝐻𝐴𝐴) = 𝑁 − 1 = 8). This means that the network is fully observable,
regarding both the conventional states (𝜃 and 𝑉) and the new ones (𝑡 and 𝑢).
A further analysis proves the system observability, by obtaining the Gain
matrix and performing the triangular factorization, as follows.
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝑈 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝑡24
𝑡25
𝑡34
𝑡35 [ 2.65 −1.51 −1.13 0.38 0.38 0.38 0.38
1 −1
1 −1
0.85 −0.85 −0.51 0.68 −0.51 0.68
0
1.61 0.74 0.37 0.12
1.36 −0.06 0.23
1.21 0.14
0.56]
. (49)
Only one zero pivot is found, at position (5, 5) referring to bus 5, what means
that only one reference is required for the entire system.
In the topological approach, the measurement graph will be composed not
only by vertices related to network buses, but also by vertices related to the new state
60
variables. Conventional measurements are treated in the same way as presented in
the last section, though it is necessary to process pseudo measurements related to CB
status (also referred as operational and structural constraints).
The measurement graph will be composed of two measurement graphs: the
first one related to buses/nodes (𝐺𝑀_𝜃), and the second related to power flows through
switching branches (𝐺𝑀_𝑡). Figure 16 depicts the test system and the measurement
graph takes the following form, as shown in Figure 17.
Figure 17 – Generalized measurement graph
The conventional measurement 𝑃1−4 only connects the vertices related to
buses 1,4. The injections measurement 𝑃1 can connect the vertices related to buses
1,4, and 5, but only one edge can be used for observability purposes. In such a case,
the edge connecting vertices 1 and 5 must be chosen. The flow measurement on
switching branch 2-4 belongs to measurement graph 𝐺𝑀_𝑡. and it connects the vertex
𝑡24 directly to the reference node 𝑡0.
Operational constraints of closed CB belong to 𝐺𝑀_𝜃 graph, and operational
constraints of open CB take the same form as flow measurements through switching
branches. In its turn, structural constraints of null injections can be used to connect
conventional states and generalized ones. In such example 𝑝4 can be used to either
connect 𝑡24 - 𝑡34 or 𝜃1 - 𝜃4. Equally, 𝑝5 is used to either connect the vertices related to
61
states 𝑡25 - 𝑡35 or buses 𝜃1 - 𝜃5. Nonetheless, for such circumstances is preferable to
use them to connect the generalized states 𝑡24 - 𝑡34, and 𝑡25 - 𝑡35, respectively.
The structural constraint of angular reference connects both measurement
graphs, enabling the entire network observable and providing a reference for the
system. At least one voltage measurement magnitude per observable island is
necessary as it plays the same role of the angular reference for the reactive part.
By interpreting the generalized graph in Figure 17, it is possible to analyze
measurement/constraint criticality. The operational constraints of closed circuit breaker
are critical, since its loss would lead to system unobservability. The same aplies for
operational constraint of open CB 𝑡25, and structural constraints 𝜃1 and 𝑝5. The
remaining measurements/constraints are all redundant.
3.2.2 Determining Observable Islands in the Generalized Approach
Figure 16 demonstrated a power network fully observable. However, if the
network is unobservable, further analysis must be carried out to find both observable
islands and unobservable branches.
Taking the example of Figure 16, and assuming the flow measurement on
switching branch 2-4 is lost and so does the power injection measurement at bus 2,
and also considering the CB 2-5 status as unknown, the Jacobian matrix takes the
following form.
62
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝐻𝐴𝐴 =
𝑃1−4
𝑃1
𝑃4
𝑝
𝑃5
𝑝
∆𝜃2−4
𝑝
∆𝜃3−5
𝑝
𝑡3−4
𝑝 [
1 −1
2 −1 −1
−1 1 −1 −1
−1 1 −1 −1
1 −1
1 −1
1 ]
. (50)
As a result, the rank of the matrix (50) is 7, which renders the network unobservable.
By forming the Gain matrix and by computing the triangular factorization, the
following 𝑈 factor is obtained.
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝑈 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝑡24
𝑡25
𝑡34
𝑡35 [ 2.65 −1.51 −1.13 0.38 0.38 0.38 0.38
1 −1
1 −1
0.85 −0.85 −0.51 0.68 −0.51 0.68
0
0.77 0.26 0.77 0.26
0.58 0.58
1
0 ]
. (51)
Two zero pivots are found, at positions (5,5) and (9,9), corresponding to bus
5 and switching branch 3-5. Such finding means the system needs two references, an
angle and a power flow. Reference flow measurements are associated with switching
branches state variables; in the same way, angular references are attached to network
islands (MONTICELLI, 1993b).
By changing the zero pivots by one, and the respective elements of the vector
𝑡𝐴, the estimated state 𝜃 is evaluated. Then, the unobservable branches are found as
follows.
63
𝑃𝑏 = 𝐴�̂� =
𝑏1−4
𝑏1−5
𝑠2−4
𝑠2−5
𝑠3−4
𝑠3−5 [ 1 −11 −1
11
11 ]
[
000000
−101 ]
=
[
000
−101 ]
.
(52)
where:
𝑠𝑘−𝑙: refers to a switching branch from generic node 𝑘 to 𝑙.
This first analysis reveals that switching branches 2-5 and 3-5 are
unobservable, and as a consequence, the pseudo-injection measurement at node 5 is
irrelevant. When the Jacobian matrix 𝐻𝐴𝐴 is updated by taking out the row
corresponding the pseudo injection measurement at node 5 and applying the triangular
factorization again, it gives the following 𝑈 factor.
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝑈 =
𝜃1
𝜃2
𝜃3
𝜃4
𝜃5
𝑡24
𝑡25
𝑡34
𝑡35 [ 2.45 −1.63 −0.82 0.41 0.41
1 −1
1 −1
0.58 −0.58 −0.58 −0.58
0
0.71 0.71
0
1
0 ]
. (53)
Accordingly, three zero pivots are found at (5,5), (7,7), and (9,9). Following
the same procedure, the flows are obtained as follows.
64
𝑃𝑏 = 𝐴�̂� =
𝑏1−4
𝑏1−5
𝑠2−4
𝑠3−4
[
1 −11 −1
11
]
[ 000000102]
= [
0000
].
(54)
From the above, one can notice that if the flows are null, all the remaining
branches and switching branches are observable. In such case, the conventional
branches from bus 1 to 4 and 5, and switching branches 2-4 and 3-4 form an
observable island. Switching branches 2-5 and 3-5 are unobservable and form two
unobservable islands.
3.3 SUMMARY OF THE CHAPTER
This chapter have represented the observability methods for power systems
modeled at the bus-branch and bus-section levels. The numerical and topological
approaches were demonstrated, and as consequence it has been shown how a power
network can be found observable, by determining the rank of Jacobian matrix and by
finding and Observable Spanning Tree. In case of unobservability, it was demonstrated
how to find unobservable branches, observable islands and irrelevant measurement
by post-processing the Jacobian matrix.
For power systems modeled at the substation level, the same numerical
algorithms could be applied, as the new state variables are considered in the Jacobian
matrix. For the topological approach, the same is true, since the new states from a new
measurement graph, which also must form an Observable Spanning Tree.
The numerical algorithm unfolded here are used in the proposed method to
evaluate the system observability in the presence of PMUs, in a pre-processing step.
65
4 OBSERVABILITY AND CRITICALITY ANALYSIS FOR GENERALIZED
STATE ESTIMATION CONSIDERING PHASOR MEASUREMENTS
This chapter presents the above mentioned methodology when coping with
GPS synchronized phasor measurements in the Generalized approach of State
Estimation.
The proposed approach is an extension of the method presented in the paper
named “Observability and criticality analyses for power systems measured by phasor
measurements” (GOL; ABUR, 2013). Such paper offered observability and criticality
analysis methods for two different kinds of current phasor measurement
configurations. However, is focused only for systems modeled at the bus-branch level.
The method is described for two current phasor measurement configurations.
In the first one, named measurement configuration type-1, current phasor
measurements have a corresponding voltage phasor measurement at the sending end
bus. Thus, it allows the deployment of the traditional methods for observability and
criticality analysis as it converts the current phasor measurement into power flow
measurements. On its turn, measurement configuration type-2 is applicable in case of
loss or bad data, when current phasor measurement may be available without the
corresponding voltage phasor measurement, therefore forbidding the conversion for
power flow measurements and the use of traditional methods.
4.1 OBSERVABILITY AND CRITICALITY METHODS FOR MEASUREMENT
CONFIGURATION TYPE-1
In general, one of the PMU’s channels measures the voltage phasor of its bus
while the remaining ones measure the current phasors of connected branches, as
depicted in Figure 18.
66
Figure 18 – Bus and branches measured by a PMU
With both voltage and current measurements available, it is possible to
compute the active and reactive power flows in those branches, turning the current
phasor measurements into power flow ones, as follows:
𝑉𝑘𝐼𝑘𝑚∗ = 𝑃𝑘𝑚 + 𝑗𝑄𝑘𝑚 (55)
In the example of Figure 18, all the current phasor measurement can be
converted into power flow ones by using the Eq. (55). This is a common practice in
industry (ATANACKOVIC et al., 2008) and it permits the use of the traditional
observability and criticality methods.
In order to show these properties for GSE, the 3 bus/5 nodes system shown
in Figure 19 is used as an example.
67
Figure 19 – 3 bus / 5 nodes system – Example 1
In this case, there is a PMU at the busbar 2, where one channel measures the
voltage phasor, and the other two channels measure the current phasors on the
switching branches 2 – 4 and 2 – 5. The actual status of the circuit breaker 2 – 5 is
opened, but unknown for purposes of demonstration.
The current phasor measurements have a voltage measurement at the
corresponding bus, so that they can be converted into power flow measurements. The
current phasor measurements, however, are placed on switching branches, where the
equation takes the following form for conversion:
𝑉𝑘𝐼𝑘𝑙∗ = 𝑡𝑘𝑙 + 𝑗𝑢𝑘𝑙 (56)
Therefore, the current phasor measurements are converted as measurements
of the generalized states to be estimated, being usable directly in the Jacobian matrix,
which takes the following form:
68
𝜃1 𝜃2 𝜃3 𝜃4 𝜃5 𝑡24 𝑡25 𝑡34 𝑡35
𝐻𝐴𝐴 =
𝑃1−4
𝑃1
𝐼2−4𝑃𝑀𝑈/𝑡
2−4
𝐼2−5𝑃𝑀𝑈/𝑡
2−5
𝜃1𝑃𝑀𝑈
𝜃2𝑃𝑀𝑈
𝑃4
𝑝
𝑃5
𝑝
∆𝜃2−4
𝑝
∆𝜃3−5
𝑝
𝑡3−4
𝑝 [
1 −1
2 −1 −1
1
1
1
1
−1 1 −1 −1
−1 1 −1 −1
1 −1
1 −1
1 ]
.
(57)
The voltage phasor measurements are also used as measurements of the
states, and in the corresponding column is represented by “1”.
Since circuit breaker 2-5 status is unknown, its operational constraint, or
pseudo measurement (∆𝜃2−5𝑝 ), is not included in the Jacobian matrix.
The matrix 𝐻𝐴𝐴 in Eq. (57) has a full rank, rendering the network observable.
When the triangular factorization is performed, none zero pivots are found. Figure 20
shows the measurement graph of the system, which yields a spanning tree.
Figure 20 – Measurement graph of the 3 bus / 5 nodes system – Example 1
69
The criticality analysis can be carried out using the traditional method, as
described in Gol and Abur (2013). The Sensitivity matrix (𝑆) is computed as described
in Eq. (58).
𝑆 = 𝐼 − 𝐻𝐴𝐴(𝐻𝐴𝐴𝑇 𝐻𝐴𝐴)−1𝐻𝐴𝐴
𝑇 (58)
where:
𝐼: is an 𝑚x𝑚 identify matrix
The zero diagonals entries in matrix 𝑆 represent critical measurements. Since
the proposed extension includes operational and structural constraints (pseudo
measurements related to CB status and null injection nodes), which models the status
of switching branches, matrix 𝑆 (zero diagonal entries) also indicates critical
constraints. In the example case of Figure 19, two measurements and two constraints
(or pseudo measurements) are flagged critical, as follows: injection measurements at
bus 1; the current phasor measurement on switching branch 2-5; the operational
constraint of CB 3-5; and structural constraint of null injection at node 5.
This result is verified by analyzing the measurement graph in Figure 20, where
it is possible to verify that if each critical measurement and constraint is lost, an
observable spanning tree is no longer formed.
An important result that is worth being pointed out here is that the voltage
phasor measurement at busbar 2 is able to prevent criticality of an operational
constraint, and it corresponds to the closed position of circuit breaker 2-4. Furthermore,
the graph in Figure 20 indicates that the direct measure of the state 𝜃2 clearly creates
an additional connection with the reference node 𝑡𝑜 through “node” 𝜃2. Consequently,
this PMU also avoids the criticality of phasor measurement at bus 1, claiming the
relevance of analyzing the observability and criticality of systems with PMU at the bus-
section level, as proposed in this thesis.
70
4.2 OBSERVABILITY AND CRITICALITY METHODS FOR MEASUREMENT
CONFIGURATION TYPE -2
The case of configuration measurement type-2 can be illustrated with the
same example of Figure 19, but removing the voltage phasor measurement at busbar
2, as shown in Figure 21.
Figure 21 - 3 bus / 5 nodes system – Example 2
In such a case, the current phasor measurements cannot be decoupled, and
the traditional approach cannot be carried out. The proposed method in Gol and Abur
(2013), extended here to process switching branch states, consists of the following
steps:
1. Disregard all the phasor measurement and process the conventional measurement
making use of the traditional method;
2. If the system is found not observable, the observable islands should be found and
each of them must be considered as super-nodes;
3. Place the non-processed phasor measurements in the simplified system consisted
of super-nodes, and check the observability again.
71
In step 1, by removing the voltage phasor measurement at bus 1 and current
phasor measurements on switch branches 2-4 and 2-5, the system is processed only
with the conventional measurements. The same system is found unobservable. After
that, step 2, applies the method shown in chapter 3, and it forms the observable
islands. The resulting system is exemplified in Figure 22.
Figure 22 - Measurement graph of the 3 bus / 5 nodes system disregarding PMU – First Step – Example 2
Bearing the picture in mind, the super-node 1 is formed by all the conventional
states and switching branches 𝑡34 and 𝑡24. Super-nodes 2 and 3 are formed by
switching branch states 𝑡25 and 𝑡35, respectively. The structural constraint of null
injection at bus 5 is a boundary injection, flagged as irrelevant. As mentioned before,
the phasor measurement at bus 1 is not being considered in this first analysis, though
an angular reference is adopted at this stage.
Step 3 processes phasor measurements on the new system, which is made
up by super-nodes. Figure 23 illustrates such system:
72
Figure 23 – Subsystem formed with Super-nodes of PMU – Example 2
In the figure above, the voltage phasor measurement at bus 1 is represented
by a phasor at super-node 1. Current phasor measurement on switching branch 2-4 is
inside the super-node 1. Since the state can be estimated, it corresponds to a voltage
phasor measurement in terms of observability (GOL; ABUR, 2013).
Finally, the current phasor measurement on switching branch 2-5 is also
represented as a voltage phasor, as it is placed on a switching branch, which is a state
to be estimated.
In order to numerically process the phasor measurements, an incidence matrix
relating the states to phasor measurements is formed. The matrix’s columns represent
the super-nodes and the rows the phasor measurements, as follows:
1 2 3
𝐴 =
𝐼2−4
𝐼2−5
𝑉1
[
1
1
1
].
(59)
Voltage phasor at bus 1 and current phasor on switching branch 2-4 are “1”s
in the column corresponding to super-node 1. Current phasor on switching branch 2-5
is a “1” in the column corresponding to super-node 2.
73
Performing the row reduced echelon form of matrix 𝐴, it provides the
identification of the anchored and floating super-nodes. In its turn, the columns
including linearly independent “1”s stand for anchored super-nodes (GOL; ABUR,
2013). Eq. (60) elucidates such identification:
1 2 3 1 2 3
𝐼2−4
𝐼2−5
𝜃1 [ 1
1
1 ]
⇒
[ 1
1
]
.
(60)
In example 2, the super-nodes 1 and 2 form an anchored super-node and the
super-node 3 is a floating super-node. After processing the phasor measurements, the
system is still identified as unobservable.
On the other hand, the structural constraint of null injection at node 5, which
was flagged irrelevant in the first step of the proposed analysis, must be considered to
finalize the process. The structural constraint at node 5 ensures the system
observability, as shown in Figure 24.
Figure 24 – Anchored and floating super-nodes – Example 2
This boundary injection/constraint allows the connections of super-nodes 2
and 3, as shown in the dotted line in Figure 22. This constraint was flagged irrelevant
in the first step of the analysis; however, after considering the phasor measurements,
74
it must be reconsidered to finalize the observability analysis. Figure 25 depicts this
situation in a graph illustration.
Figure 25 - Measurement graph of the 3 bus / 5 nodes system disregarding PMU –Effect of irrelevant injection measurement/constraint– Example 2
Another possible case is when an irrelevant boundary injection is not
necessary to render the system observable. In its turn, it allows the use of Sensitivity
matrix to find critical measurements, as discussed ahead. Consider the new situation
depicted in Figure 26.
75
Figure 26 - 3 bus / 5 nodes system – Example 3
Conventional measurements process reveals the same results from the
previous example, and it has the same super-nodes shown in Figure 23. Super node
3, however, also has a voltage phasor measurement associated to it, corresponding to
the current phasor on switching branch 3-5, as shown in Figure 27.
Figure 27 - Subsystem formed with Super-nodes – Example 3
The 𝐴 matrix and its corresponding row reduced echelon form take the
following form, as demonstrated in Eq. (61).
76
1 2 3 1 2 3
𝐴 =
𝐼2−4
𝐼2−5
𝐼3−5
𝜃1 [
1
1
1
1 ]
⇒
[
1
1
1
]
.
(61)
The row reduced echelon form shows that the system is only found observable
when phasor measurements are taken into account, by having “1”s at each linearly
independent column of the matrix. Once the system is observable, it is possible to use
the 𝐴 matrix to compute the Sensitivity matrix 𝑆, as demonstrated in Eq. (62) (GOL;
ABUR, 2013).
𝑆 = 𝐼 − 𝐴(𝐴𝑇𝐴)−1𝐴𝑇 (62)
The null elements in the diagonal of the matrix 𝑆 correspond to critical phasor
measurements.
In the presence of irrelevant injection, however, it is necessary to go back one
step and consider the boundary injection/constraint. This is the case of Example 3, in
which the effect of the structural constraint of null injection at node 5 was not
considered yet. Considering such example, the boundary injection connects the super-
nodes 2 and 3, which corresponds to switching branches 2-5 and 3-5, as shown in
Figure 22. Thus, at this stage, only two super-nodes must be formed to analyze the
effect of phasor measurements, as outlined in Figure 28.
77
Figure 28- Subsystem formed with Super-nodes, considering the boundary injection – Example 3
Therefore, the actual 𝐴 matrix, its corresponding row reduced echelon form,
and the Sensitivity matrix 𝑆 take the following forms.
1 2 1 2
𝐴 =
𝐼2−4
𝐼2−5
𝐼3−5
𝜃1 [
1
1
1
1 ]
⇒
[
1
1
]
(63)
𝑆 =
𝐼2−4
𝐼2−5
𝐼3−5
𝜃1 [
0.5 −0.5
0.5 −0.5
−0.5 0.5
−0.5 0.5 ]
. (64)
From those, none of the phasor measurements are flagged critical.
This numerical criticality analysis only allows the identification of critical
phasor measurements. The critical conventional measurements and constraints can
be identified by applying the conventional analysis separately for each super-node.
The step-by-step procedure of the proposed algorithm is presented as follows:
78
1 Disregard all phasor measurements and perform the Traditional
Observability Analysis (TOA);
2 If the system is Observable, proceed to step 12; otherwise find the
observable islands and form super-nodes;
3 Set the phasor measurements in the corresponding simplified system
composed by super-nodes; form the A matrix, and find its echelon reduced
form;
4 Check the observability by looking if there are “1”s in all columns, which are
at linearly independent rows. Despite of the result, also check if there is/are
irrelevant measurements/constraints;
5 If the system is observable and there is/are irrelevant
measurements/constraints proceed to step 6; if the system is observable and
there are no irrelevant measurements/constraints proceed to step 8; if the
system is unobservable and there is/are irrelevant
measurements/constraints proceed to step 9; otherwise, proceed to step 11;
6 Find all connections among super-nodes that might be provided by irrelevant
measurements/constraints and merge them, thereby structuring a new
simplified system;
7 Set the phasor measurements in the new subsystem, and form the A matrix;
8 Form the S matrix with A matrix; find critical phasor measurements and
proceed to step 12;
9 Find the anchored and floating super-nodes, form a new simplified system,
and process the irrelevant measurements/constraints;
10 If the irrelevant measurements render the system observable proceed to
step 12; otherwise, proceed to step 11;
11 Find observable islands, unobservable branches and stop the analysis;
12 Proceed for SE.
The proposed algorithm can be summarized in the flow chart in Figure 29. The
Highlighted boxes refer to the contribution of this work.
79
Figure 29 – Algorithm Flowchart for Observability and Criticality Analysis for Measurements Type-2
80
4.3 SUMMARY OF THE CHAPTER
This chapter presented the proposed method to treat phasor measurements
for systems modeled at the substation level, considering two kinds of measurement
configuration. In the first one, traditional methods could be applied as current phasor
measurements were converted into power flow measurements. In the second case,
however, phasor measurements were processed after the conventional
measurements, in a simplified system of super-nodes.
The impact of irrelevant measurements was also discussed and it was able to
show their capability to render a system fully observable when phasor measurements
are considered. Along with that, the proposed algorithm was presented in a step-by-
step procedure as well as summarized in a flow chart.
Subsequently, it was pointed out how phasor measurements present
themselves as a great advantage to be used in GSE. For such cases, the operational
constraint of closed CB is no longer critical, providing a redundancy for logical
measurements.
81
5 TESTS AND RESULTS ANALYSIS
This chapter presents the results of the developed algorithm, which was
implemented in MATLAB. The Power Education Toolbox (PET) software (ABUR;
MAGNAGO; KRIZAN, 2014) was also used as a means to simplify the way the system
is modelled, and to generate the system data and measurements. The well-known
IEEE 14 bus benchmark system was used as a base case, with three substations
modeled in detail.
The next sections of this chapter present the base case, the modeled
substations, and the different measurement designs deployed to validate the proposed
method of Observability and Criticality methods for power networks modeled at the
substation level with PMU.
5.1 BASE CASE
Figure 30 discloses three buses, 10, 11 and 14, of the IEEE 14 bus system
adopted to be modeled in detail at the substation level:
Figure 30 – IEEE 14 Bus system with Modeled Substations
82
With those substations modeled in detail, it has been assumed the status of
two circuit breakers as unknown, and an initial set of conventional measurements, as
shown in Figure 31.
Figure 31 – Substation modeled in detail – Base Case
The new buses/nodes were numerated according to an increase of the system
size, that is, from 14 buses to 23 buses/nodes. CB 15-16 status is open but unknown
by the system operator, and the status of CB 18-21 is closed and also unknown, so as
to simulate a communication failure. The unknown situation aims to simulate a situation
of communication failure, where the system operator does not know the actual CB
status. The base set of measurements, i.e. the measurement set in the system
modeled at the bus-branch level, is composed by several measurements such that no
one is critical in the bus-branch level, and it provides a clear analysis for the substation
level6.
6 For more details, please refer to Appendix B
83
Running the GSE algorithm, the system is found unobservable, as described
ahead. All the conventional states (𝜃, 𝑉) and conventional branches are observable, as
well as switching branch states (𝑡, 𝑢) of substations 10 and 14. The switching states of
substation 11 though, are unobservable, as they form four unobservable islands.
According to the proposed algorithm, the system is reduced into five super-nodes, as
it can be seen in Figure 32.
Figure 32 – Super-nodes of base case
Furthermore, the injection measurement at bus 11, and the structural
constraints of null injection at buses 15, 16, and 17 are flagged irrelevant by the
proposed approach.
5.2 CASE A – MEASUREMENT CONFIGURATION TYPE 1
The first case illustrates the use of measurement configuration type-1,
described in section 4.1 of Chapter 4. Figure 33 shows the set of measurements for
this case:
84
Figure 33 – Modeled Substations – Case Study A
In comparison to the base case, in this very one there are two voltage phasor
measurements set at busbars 10 and 11, and two current phasor measurements set
on switching branches 10-23 and 11-16. These current phasor measurements can be
converted into power flow measurements, allowing the use of traditional methods for
observability and criticality analysis.
The use of current phasor measurement on switching branch 11-16, is enough
to render the system observable when combined with the voltage phasor at the
corresponding busbar. The irrelevant injection measurements/constraints at busbars
11, 15, 16 and 17 play an important role here, since they allow the connection among
the super-nodes 2 to 5, rendering a full observability.
The criticality analysis reveals that current phasor measurement on switching
branch 11-16 is critical, as well as the operation constraints of closed CB 11-17, 15-
17, 10-23, 14-19, and 14-20.
Although the pair of phasor measurements at busbar 10 and switching branch
10-23 are redundant, they make the operational constraint of closed CB 10-21, which
was initially critical, but it turned out to be non-critical. The current phasor measurement
also makes all the operational constraints of open circuit breakers redundant as well.
85
Running the GSE algorithm, a final solution was found after 4 iterations. See
appendix B for more details.
5.3 CASE B – MEASUREMENT CONFIGURATION TYPE 2
In case B, the current phasor measurement on switching branch 11-16 is lost,
as well as the voltage phasor measurement at busbar 10, as shown in Figure 34.
Figure 34 - Modeled Substations – Case Study B
In this case, the current phasor measurement on switching branch 10-23
cannot be converted into a power flow measurement, and the traditional approach
cannot be applied.
In this situation, the proposed algorithm first processes the conventional
measurements using the traditional observability method, that is, all the phasor
measurements are disregarded in the first moment as described in Step 1 of the
proposed algorithm (please refer to Section 4.2, of Chapter 4).
At the end of Step 1, the system is reduced to the simplified system. The
results are the same as shown in subsection 5.1, and depicted in Figure 32.
86
The phasor measurement at busbar 11 directly measures a state to be
estimated (𝜃11), and current phasor measurement on switching branch 10-23 is a
function of another state to be estimated (𝑡10−23). Therefore, according to the proposed
algorithm, both measurements can be represented in the same way as voltage phasor
measurements on super-node 1, as both states are inside super-node 1, as shown in
Figure 357.
Figure 35 - Super-nodes of case study B
This measurement set is not able of rendering the system fully observable.
Both phasor measurements are redundant and they only make the conventional states
and switching states of substation 10 and 14 observable. The switching states of
substation 11 remains unobservable, and as a consequence, injection measurement
at busbar 11 and the structural constraints of null injection at nodes 15, 16 and 17
remains irrelevant.
7 The third voltage phasor at super-node 1 is a voltage phasor measurement placed at bus 1
87
Running the GSE algorithm, a final solution was not found, since it does not
converge due to system unobservability.
5.4 CASE C – MEASUREMENT CONFIGURATION TYPE 2
In the case C, a current phasor measurement is placed on switching branch
15-16, as exhibited in Figure 36.
Figure 36 - Modeled Substations – Case Study C
This new current phasor measurement cannot be converted into power flow
measurement, since busbar 15 has no voltage phasor measurement. Following the
proposed algorithm, this current phasor is represented as a voltage phasor on super-
node 2, and it corresponds to switching states 𝑡15−16, as shown in Figure 37.
88
Figure 37 - Super-nodes of case study C
Now, the super-nodes 1 and 2 form an anchored super-node, and those
remained form three floating super-nodes, as shown in Figure 38.
Figure 38 – Anchored and Floating super-nodes – Case C
At this point, the system is unobservable. However, there are injection
measurements/constraints (𝑃11𝑚, 𝑃15
𝑐 , 𝑃16𝑐 , and 𝑃17
𝑐 ) which were flagged irrelevant and,
89
according to the proposed algorithm, they should be re-evaluated at this point (see
Step 9 of the algorithm). Figure 39 offers the possible connections.
Figure 39 - Anchored and Floating super-nodes and injection measurement connections – Case C
It can be noticed that the injection measurement at busbar 11 connects super-
nodes 4 and 5, and the structural constraint of null injection at node 17 connects super-
nodes 3 and 5. The remaining structural constraints of null injection at nodes 15 and
16 connect the super-node 3 and 4 with super-node, rendering the system fully
observable.
Running the GSE algorithm, a final solution was found after 5 iterations. See
appendix B for more details.
5.5 CASE D – MEASUREMENT CONFIGURATION TYPE 2
In the case D, all the switching branches of substation 11 are measured by
current phasor measurements, though without a voltage phasor one, as portrayed in
Figure 40.
90
Figure 40 - Modeled Substation – Case Study D
In this case, all the current phasor measurements in substation 11 render
observability for all switching states, which were unobservable in the base case. The
current phasor measurements are represented in the proposed approach in the same
way as voltage phasor in super-nodes 2 to 5, as depicted in Figure 41.
Figure 41 - Super-nodes of case study D
Since the phasor measurements render the system fully observable by making
no use of irrelevant measurements, an application of the criticality analysis using the
91
matrix 𝐴, is suitable. The irrelevant measurements, in its turn, must be considered
before such analysis is carried out, as described in Step 4 of the algorithm.
Step 6 points out the procedure for this case, that is, to find all connections
among super-nodes provided by irrelevant measurements and merge them so as to
form a new simplified system. The result of this analysis indicates that super-nodes 2
to 5 are all connected, as in Figure 42.
Figure 42 - Super-nodes of case study D with irrelevant measurements and constraints
Therefore, the phasor measurements are processed considering two super-
nodes when it comes to criticality analysis, as detailed in Figure 43.
Figure 43 – Super-node of case study D for Criticality Analysis
92
Matrix 𝐴 is assembled considering only two super-nodes and six phasor
measurements (see Step 7 of the algorithm), as follows:
1 2
𝐴 =
𝜃1
𝐼10−23
𝐼11−16
𝐼11−17
𝐼15−16
𝐼15−17 [
1
1
1
1
1
1 ]
.
(65)
Therefore, the Sensitivity matrix 𝑆𝐴 (determined in Step 8) takes the following
form:
𝑆𝐴 =
𝜃1
𝐼10−23
𝐼11−16
𝐼11−17
𝐼15−16
𝐼15−17 [
0.5 −0.5
−0.5 0.5
0.75 −0.25 −0.25 −0.25
−0.25 0.75 −0.25 −0.25
−0.25 −0.25 0.75 −0.25
−0.25 −0.25 −0.25 0.75 ]
. (66)
As it has not been found any zero elements in the 𝑆𝐴 matrix, none of the phasor
measurements are flagged critical.
Running the GSE algorithm, a final solution was found after 5 iterations. See
appendix B for more details.
93
5.6 CASE E – ISLAND NODE
In case E, the status of CB 10-21 is unknown, causing the islanding of nodes
10 and 23, as seen in Figure 44.
Figure 44 – Modeled substation – Case E
For such circumstances, the traditional numerical algorithm was not capable
of identifying the observable and unobservable portions of the system, and as a
consequence was not able to form the super-nodes. Since it failed to provide a
satisfactory result in this first step, this case could not be fully investigated in the light
of the proposed method. Although the analysis could not be carried out, this case
serves to highlight the impact of an operational constraint of closed CB.
94
5.7 SUMMARY OF THE CHAPTER
This chapter presented the results of the proposed method in systems
modeled at the substation level. A base case was disclosed whose composition
consisted of only conventional measurements and two unknown CB status, which
rendered the system unobservable. Thus, four situations combining measurements
configurations type 1 and 2 were presented for analysis. The results have shown that
the method could be extended for systems in substation level. Moreover, it is worth
mentioning that all the results were validated by using the Topological approach,
proposed in Simões Costa, Lourenço and Clements (2002), and also by running the
developed GSE algorithm. The results achieved by the developed GSE algorithm are
thoroughly presented in Appendix B.
95
6 CONCLUDING REMARK AND FUTURE STUDY
This chapter presents the concluding remarks of this work by reassessing the
proposed objectives, singling out the achievements and main results. In addition, it
also presents suggestions for future studies.
6.1 CONCLUDING REMARKS
This Master’s thesis presented the theory of the well-known WLS Traditional
and Generalized State Estimation related to conventional measurements. The
inclusion of synchronized phasor measurements was discussed, addressing the use
of voltage and current phasor measurements in the bus-branch and bus-section levels.
The equations used in two different approaches of the GSE with phasor measurements
were derived, providing an easy way to implement them, and providing satisfactory
results once included in the already developed GSE algorithm.
The theory of observability analysis, including the topological and numerical
approaches were also presented. The numerical approach was deeply investigated as
its use was paramount for the proposed method, since it was used to identify
observable and unobservable portions of the system, forming the super-nodes in the
proposed methods.
The proposed methods of Observability and Criticality Analysis were based
on the work of Gol and Abur (2013), which have suggested two approaches for dealing
with different measurement configurations for systems modeled at the bus-branch
model. In the same way, the proposed methods were implemented and tested for those
two measurement configurations, though taking into account the system modeled at
substation level.
In addition to that, an algorithm was developed in MATLAB and it was
validated via tests over an IEEE system. The results have revealed that the
observability method could be extended for substation level, enabling a determination
whether the system is observable or not, as well as unobservable switching branches.
96
Notwithstanding, a specific situation of failure was pointed out in case of node islanding
when an operational constraint of closed CB is unknown.
Additionally, the measurement criticality analysis presented some drawbacks,
since it was not possible to make use of it in cases where irrelevant injection
measurements/constraints had been used to render the system observability.
Besides the drawbacks found, the results have shown a great advantage of
using phasor measurements inside the substations. The PMUs were capable of
providing a greater level of redundancy for operational constraints. It was observed
that critical constraints of closed CB were no longer critical after the usage of voltage
phasor measurements in some busbars. Current phasor measurements also provided
more redundancy for operational constraints of opened CB, since they directly
measure the state to be estimated.
6.2 FURTHER STUDY
This work has acknowledged the applicability of Observability and Criticality
methods for power systems modeled at the bus-section level. However, the proposed
method of measurement criticality for measurements in configuration type-2 was
applicable only for cases where irrelevant injection measurements had not been used
to ensure the system observability. As a further study, this method can be improved in
order to cope with such situations, and provide an analysis of critical measurements
for all cases.
In addition to that, an improvement of the numerical algorithm to deal with
island situations would as well be developed. A way to approach it would be pre-
processing the system, separating the island portions from the rest, to apply the
proposed method afterwards. Another way would be using a hybrid algorithms
leveraging the benefits of the topological approach.
The proposed method of measurement criticality analysis could also be
attached in the objective function of an optimization algorithm, aiming to find an optimal
design of PMUs allocation, and avoiding critical measurements and operational
constraints.
97
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APPENDIX A – POWER FLOW AND INJECTION EQUATIONS
This appendix presents all the referred equations cited in the previous
chapters.
(A) Power Flow Equations
Considering the unified branch model (MONTICELLI, 1999) presented at
Figure 45, in which there are two phase shifting transformers, one of each side of the
line, it is possible to derive all the power flow equations.
Figure 45 – Unified branch Model Source: Monticelli (1999)
Active and reactive power flows from bus 𝑘 to bus 𝑚:
𝑃𝑘𝑚 = |𝑎𝑘𝑚|2𝑉𝑘2𝑔𝑘𝑚 − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚( 𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)
+ 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) )
𝑄𝑘𝑚 = −|𝑎𝑘𝑚|2𝑉𝑘2(𝑏𝑘𝑚 + 𝑏𝑘𝑚
𝑠ℎ ) − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚( 𝑔𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)
− 𝑏𝑘𝑚𝑐𝑜𝑠(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) )
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where:
𝑎𝑘𝑚 = |𝑎𝑘𝑚|𝑒𝑗𝜑𝑘𝑚;
𝑡𝑘𝑚 = |𝑡𝑚𝑘|𝑒𝑗𝜑𝑚𝑘;
𝑔𝑘𝑚: is the series conductance;
𝑏𝑘𝑚: is the series susceptance.
Active and reactive power flows from bus 𝑚 to bus 𝑘:
𝑃𝑚𝑘 = |𝑡𝑚𝑘|2𝑉𝑚
2𝑔𝑘𝑚 − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚( 𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑚𝑘 − 𝜑𝑘𝑚)
+ 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑚𝑘 − 𝜑𝑘𝑚) )
𝑄𝑚𝑘 = −|𝑡𝑚𝑘|2𝑉𝑚
2(𝑏𝑚𝑘 + 𝑏𝑚𝑘𝑠ℎ ) − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚( 𝑔𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑚𝑘 − 𝜑𝑘𝑚)
− 𝑏𝑘𝑚𝑐𝑜𝑠(𝜃𝑘𝑚 + 𝜑𝑚𝑘 − 𝜑𝑘𝑚) )
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(B) Power Injection Equations
By only given a bus 𝑘, the power injection equations can also be derived:
Figure 46 – Generic Bus Source: Monticelli (1999)
By applying Kirchholff´s Current Law (KCL) yields:
𝑃𝑘 = 𝑉𝑘 ∑ 𝑉𝑗(𝐺𝑘𝑚 cos 𝜃𝑘𝑚 + 𝐵𝑘𝑚 sin 𝜃𝑘𝑚)
𝑚∈𝑁𝑘
𝑄𝑘 = 𝑉𝑘 ∑ 𝑉𝑚(𝐺𝑘𝑚 sin 𝜃𝑘𝑚 − 𝐵𝑘𝑚 cos 𝜃𝑘𝑚)
𝑚∈𝑁𝑘
where:
𝐺𝑘𝑚 and 𝐵𝑘𝑚: are the real and imaginary parts of the admittance nodal matrix;
𝑁𝑘: is the set of buses connected to bus 𝑘.
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(C) Jacobian Matrix Equations
Once the power flows and injection equations were derived, the Jacobian
matrix equations are given as follow.
𝜕𝑃𝑘𝑚
𝜕𝜃𝑘= |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚(𝑔𝑘𝑚 sin( 𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)𝑏𝑘𝑚 − cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑃𝑘𝑚
𝜕𝜃𝑚= −|𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚(𝑔𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) − 𝑏𝑘𝑚(cos ( 𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)))
𝜕𝑃𝑘𝑚
𝜕𝑉𝑘= 2|𝑎𝑘𝑚|2𝑔𝑘𝑚𝑉𝑘 − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑚(𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)
+ 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑃𝑘𝑚
𝜕𝑉𝑚= −|𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘(𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) + 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑄𝑘𝑚
𝜕𝜃𝑘= −|𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚(𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) + 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑄𝑘𝑚
𝜕𝜃𝑚= |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘𝑉𝑚(𝑔𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) + 𝑏𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑄𝑘𝑚
𝜕𝑉𝑘= −2|𝑎𝑘𝑚|2(𝑏𝑘𝑚 + 𝑏𝑘𝑚
𝑠ℎ )𝑉𝑘 − |𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑗(𝑔𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘)
− 𝑏𝑘𝑚 cos(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
𝜕𝑄𝑘𝑚
𝜕𝑉𝑚= −|𝑎𝑘𝑚||𝑡𝑚𝑘|𝑉𝑘(𝑔𝑘𝑚 sin(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘) − 𝑏𝑘𝑚𝑐𝑜𝑠(𝜃𝑘𝑚 + 𝜑𝑘𝑚 − 𝜑𝑚𝑘))
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𝜕𝑃𝑘
𝜕𝜃𝑘= −𝑉𝑘
2𝐵𝑘𝑘 + ∑ 𝑉𝑘𝑉𝑚(−𝐺𝑘𝑚 sen 𝜃𝑘𝑚 + 𝐵𝑘𝑚 cos 𝜃𝑘𝑚)
𝑁
𝑚=1
𝜕𝑃𝑘
𝜕𝜃𝑚= 𝑉𝑘𝑉𝑚(𝐺𝑘𝑚 sen 𝜃𝑘𝑚 − 𝐵𝑘𝑚 cos 𝜃𝑘𝑚)
𝜕𝑃𝑘
𝜕𝑉𝑘= 𝑉𝑘𝐺𝑘𝑘 + ∑ 𝑉𝑚(𝐺𝑘𝑚 cos 𝜃𝑘𝑚 + 𝐵𝑘𝑚 sen 𝜃𝑘𝑚)
𝑁
𝑚=1
𝜕𝑘
𝜕𝑉𝑚= 𝑉𝑘(𝐺𝑘𝑚 cos 𝜃𝑘𝑚 + 𝐵𝑘𝑚 sen 𝜃𝑘𝑚)
𝜕𝑄𝑘
𝜕𝜃𝑘= −𝑉𝑘
2𝐺𝑘𝑘 + ∑ 𝑉𝑘𝑉𝑚(𝐺𝑘𝑚 cos 𝜃𝑘𝑚 + 𝐵𝑘𝑚 sen𝜃𝑘𝑚)
𝑁
𝑚=1
𝜕𝑄𝑘
𝜕𝜃𝑚= 𝑉𝑘𝑉𝑚(−𝐺𝑘𝑚 cos 𝜃𝑘𝑚 − 𝐵𝑘𝑚 sen 𝜃𝑘𝑚)
𝜕𝑄𝑘
𝜕𝑉𝑘= −𝑉𝑘𝐵𝑘𝑘 + ∑ 𝑉𝑚(𝐺𝑘𝑚 sen 𝜃𝑘𝑚 − 𝐵𝑘𝑚 cos𝜃𝑘𝑚)
𝑁
𝑚=1
𝜕𝑄𝑘
𝜕𝑉𝑚= 𝑉𝑘(𝐺𝑘𝑚 sen 𝜃𝑘𝑚 − 𝐵𝑘𝑚 cos 𝜃𝑘𝑚)
Voltage magnitude measurements
𝜕𝑉𝑘
𝜕𝜃𝑘= 0,
𝜕𝑉𝑘
𝜕𝜃𝑚= 0,
𝜕𝑉𝑘
𝜕𝑉𝑘= 1,
𝜕𝑉𝑘
𝜕𝑉𝑚= 0
107
APPENDIX B – SYSTEM DETAILS AND RESULTS OF GSE ALGORITHM
This appendix presents a description of the modeled system and the results
from the developed GSE algorithm.
(A) System Modeling
The entire system was modeled at PET (Power Education Toolbox), with the
switching branches modeled as transmission lines with low and high impedance values
for open and closed CB, respectively. Having a few branches modeled with atypical
impedance values, the software was able to run a power flow and generate the
measurements for state estimation, thereby providing similar results compared to the
system modeled at the bus-branch level. Figure 47 shows the entire system modeled
in PET, with the conventional measurements spread all over it.
Figure 47 – IEEE 14 bus system modeled in PET
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The double voltage magnitude measurement at buses B1, B10, and B11
stands for a voltage magnitude and phasor measurements. These last, in their turn,
were emulated by converting some power flow measurements into current ones.
The measurements were generated by a power flow algorithm, with a
mismatch of 1x10-3. Gaussian noise was added to all measurements in order to
emulate a real condition. Conventional measurements were considered with standard
deviations of 1x10-3 (power flow and injections) and 1x10-4 for phasor measurements
(voltage and current). The State Estimation was performed with a mismatch of 1x10-5.
(B) Results from GSE Algorithm
After carrying out the observability and measurement criticality analyses,
presented in chapter 5, the developed GSE algorithm was run for all cases in order to
check if the state estimation process was possible and the results are shown in the
form of screen shots of the outputs from the algorithm.
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110
111
112
113
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Only the results of test cases A, C, and D are shown here, as they refer to
observable systems. The GSE algorithm did not converge for test cases B and E.